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<div class="section" id="fwk-redden-ch04_s08" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">4.8</span> Applications and Variation</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch04_s08_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch04_s08_o01" numeration="arabic">
<li>Solve applications involving uniform motion (distance problems).</li>
<li>Solve work-rate applications.</li>
<li>Set up and solve applications involving direct, inverse, and joint variation.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch04_s08_s01" version="5.0" lang="en">
<h2 class="title editable block">Solving Uniform Motion Problems</h2>
<p class="para block" id="fwk-redden-ch04_s08_s01_p01"><span class="margin_term"><a class="glossterm">Uniform motion (or distance)</a><span class="glossdef">Described by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2437" display="inline"><mrow><mi>D</mi><mo>=</mo><mi>r</mi><mi>t</mi></mrow></math></span>, where the distance <em class="emphasis">D</em> is given as the product of the average rate <em class="emphasis">r</em> and the time <em class="emphasis">t</em> traveled at that rate.</span></span> problems involve the formula <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2438" display="inline"><mrow><mi>D</mi><mo>=</mo><mi>r</mi><mi>t</mi></mrow></math></span>, where the distance <em class="emphasis">D</em> is given as the product of the average rate <em class="emphasis">r</em> and the time <em class="emphasis">t</em> traveled at that rate. If we divide both sides by the average rate <em class="emphasis">r</em>, then we obtain the formula</p>
<p class="para block" id="fwk-redden-ch04_s08_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2439" display="block"><mrow><mi>t</mi><mo>=</mo><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s08_s01_p03">For this reason, when the unknown quantity is time, the algebraic setup for distance problems often results in a rational equation. We begin any uniform motion problem by first organizing our data with a chart. Use this information to set up an algebraic equation that models the application.</p>
<div class="callout block" id="fwk-redden-ch04_s08_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch04_s08_s01_p04">Sally traveled 15 miles on the bus and then another 72 miles on a train. The train was 18 miles per hour faster than the bus, and the total trip took 2 hours. What was the average speed of the train?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p05">First, identify the unknown quantity and organize the data.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p06">Let <em class="emphasis">x</em> represent the average speed (in miles per hour) of the bus.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p07">Let <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2440" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>18</mn></mrow></math></span> represent the average speed of the train.</p>
<div class="informalfigure large">
<img src="section_07/b7413969d641349eb755311c73c61177.png">
</div>
<p class="para" id="fwk-redden-ch04_s08_s01_p09">To avoid introducing two more variables for the time column, use the formula <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2441" display="inline"><mrow><mi>t</mi><mo>=</mo><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow><mo>.</mo></math></span> The time for each leg of the trip is calculated as follows:</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2442" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>p</mi><mi>e</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>o</mi><mi>n</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mi> </mi><mi>b</mi><mi>u</mi><mi>s</mi><mo>:</mo></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mi>t</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>15</mn></mrow><mi>x</mi></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>p</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi> </mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>r</mi><mi>a</mi><mi>i</mi><mi>n</mi><mo>:</mo></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mi>t</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>72</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>18</mn></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s01_p11">Use these expressions to complete the chart.</p>
<div class="informalfigure large">
<img src="section_07/571e71edc4fd9296502b0ed308cda7bf.png">
</div>
<p class="para" id="fwk-redden-ch04_s08_s01_p13">The algebraic setup is defined by the time column. Add the time spent on each leg of the trip to obtain a total of 2 hours:</p>
<div class="informalfigure large">
<img src="section_07/450994967a5b3df0ff3bff0ec81f854f.png">
</div>
<p class="para" id="fwk-redden-ch04_s08_s01_p15">We begin solving this equation by first multiplying both sides by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2443" display="inline"><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>18</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s01_p16"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2444" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>15</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mrow><mn>72</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>18</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>18</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>15</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mrow><mn>72</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>18</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>18</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>18</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mn>15</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>18</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mn>72</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>18</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>18</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>15</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>18</mn><mo stretchy="false">)</mo><mo>+</mo><mn>72</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>18</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>15</mn><mi>x</mi><mo>+</mo><mn>270</mn><mo>+</mo><mn>72</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>36</mn><mi>x</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>87</mn><mi>x</mi><mo>+</mo><mn>270</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>36</mn><mi>x</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>51</mn><mi>x</mi><mo>−</mo><mn>270</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s01_p17">Solve the resulting quadratic equation by factoring.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p18"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2445" display="block"><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mspace width="2em"></mspace><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>51</mn><mi>x</mi><mo>−</mo><mn>270</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>30</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>30</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>30</mn></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s01_p19">Since we are looking for an average speed we will disregard the negative answer and conclude the bus averaged 30 mph. Substitute <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2446" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>30</mn></mrow></math></span> in the expression identified as the speed of the train.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2447" display="block"><mrow><mi>x</mi><mo>+</mo><mn>18</mn><mo>=</mo><mn>30</mn><mo>+</mo><mn>18</mn><mo>=</mo><mn>48</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s01_p21">Answer: The speed of the train was 48 mph.</p>
</div>
<div class="callout block" id="fwk-redden-ch04_s08_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch04_s08_s01_p22">A boat can average 12 miles per hour in still water. On a trip downriver the boat was able to travel 29 miles with the current. On the return trip the boat was only able to travel 19 miles in the same amount of time against the current. What was the speed of the current?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p23">First, identify the unknown quantities and organize the data.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p24">Let <em class="emphasis">c</em> represent the speed of the river current.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p25">Next, organize the given data in a chart. Traveling downstream, the current will increase the speed of the boat, so it adds to the average speed of the boat. Traveling upstream, the current slows the boat, so it will subtract from the average speed of the boat.</p>
<div class="informalfigure large">
<img src="section_07/99fc375317946134df019ab47068fd8b.png">
</div>
<p class="para" id="fwk-redden-ch04_s08_s01_p27">Use the formula <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2448" display="inline"><mrow><mi>t</mi><mo>=</mo><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow></math></span> to fill in the time column.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p28"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2449" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>t</mi><mi>r</mi><mi>i</mi><mi>p</mi><mtext> </mtext><mi>d</mi><mi>o</mi><mi>w</mi><mi>n</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>r</mi><mo>:</mo></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mi>t</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>29</mn></mrow><mrow><mn>12</mn><mo>+</mo><mi>c</mi></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>t</mi><mi>r</mi><mi>i</mi><mi>p</mi><mtext> </mtext><mi>u</mi><mi>p</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>r</mi><mo>:</mo></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mi>t</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>19</mn></mrow><mrow><mn>12</mn><mo>−</mo><mi>c</mi></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="informalfigure large">
<img src="section_07/c221b40b4e6f23f5d31a99a4ac317584.png">
</div>
<p class="para" id="fwk-redden-ch04_s08_s01_p30">Because the boat traveled the same amount of time downriver as it did upriver, finish the algebraic setup by setting the expressions that represent the times equal to each other.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2450" display="block"><mrow><mfrac><mrow><mn>29</mn></mrow><mrow><mn>12</mn><mo>+</mo><mi>c</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>19</mn></mrow><mrow><mn>12</mn><mo>−</mo><mi>c</mi></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s01_p32">Since there is a single algebraic fraction on each side, we can solve this equation using cross multiplication.</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p33"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2451" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>29</mn></mrow><mrow><mn>12</mn><mo>+</mo><mi>c</mi></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>19</mn></mrow><mrow><mn>12</mn><mo>−</mo><mi>c</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>29</mn><mrow><mo>(</mo><mrow><mn>12</mn><mo>−</mo><mi>c</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>19</mn><mrow><mo>(</mo><mrow><mn>12</mn><mo>+</mo><mi>c</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>348</mn><mo>−</mo><mn>29</mn><mi>c</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>228</mn><mo>+</mo><mn>19</mn><mi>c</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>120</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>48</mn><mi>c</mi></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>120</mn></mrow><mrow><mn>48</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>c</mi></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>5</mn><mn>2</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>c</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s01_p34">Answer: The speed of the current was <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2452" display="inline"><mrow><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> miles per hour.</p>
</div>
<div class="callout block" id="fwk-redden-ch04_s08_s01_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s08_s01_p35"><strong class="emphasis bold">Try this!</strong> A jet aircraft can average 160 miles per hour in calm air. On a trip, the aircraft traveled 600 miles with a tailwind and returned the 600 miles against a headwind of the same speed. If the total round trip took 8 hours, then what was the speed of the wind?</p>
<p class="para" id="fwk-redden-ch04_s08_s01_p36">Answer: 40 miles per hour</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/0NglBthTwss" condition="http://img.youtube.com/vi/0NglBthTwss/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/0NglBthTwss" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch04_s08_s02" version="5.0" lang="en">
<h2 class="title editable block">Solving Work-Rate Problems</h2>
<p class="para editable block" id="fwk-redden-ch04_s08_s02_p01">The rate at which a task can be performed is called a <span class="margin_term"><a class="glossterm">work rate</a><span class="glossdef">The rate at which a task can be performed.</span></span>. For example, if a painter can paint a room in 6 hours, then the task is to paint the room, and we can write</p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2453" display="block"><mrow><mfrac><mrow><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext>task</mtext></mrow><mrow><mn>6</mn><mtext> </mtext><mtext>hours</mtext></mrow></mfrac><mstyle color="#007fbf"><mtext> work rate</mtext></mstyle></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p03">In other words, the painter can complete <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2454" display="inline"><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></math></span> of the task per hour. If he works for less than 6 hours, then he will perform a fraction of the task. If he works for more than 6 hours, then he can complete more than one task. For example,</p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2455" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>w</mi><mi>o</mi><mi>r</mi><mi>k</mi><mtext>-</mtext><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>×</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>a</mi><mi>m</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>t</mi><mi>a</mi><mi>s</mi><mi>k</mi><mtext> </mtext><mi>c</mi><mi>o</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>e</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow><mn>6</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>×</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>3</mn><mtext> </mtext><mtext> </mtext><mi>h</mi><mi>r</mi><mi>s</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>o</mi><mi>n</mi><mi>e</mi><mtext>-</mtext><mi>h</mi><mi>a</mi><mi>l</mi><mi>f</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi><mtext> </mtext><mi>p</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow><mn>6</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>×</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>6</mn><mtext> </mtext><mtext> </mtext><mi>h</mi><mi>r</mi><mi>s</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>1</mn><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>o</mi><mi>n</mi><mi>e</mi><mtext> </mtext><mi>w</mi><mi>h</mi><mi>o</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi><mtext> </mtext><mi>p</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow><mn>6</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>×</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>12</mn><mtext> </mtext><mtext> </mtext><mi>h</mi><mi>r</mi><mi>s</mi><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>2</mn><mtext> </mtext><mstyle color="#007fbf"><mrow><mi>t</mi><mi>w</mi><mi>o</mi><mtext> </mtext><mi>w</mi><mi>h</mi><mi>o</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi><mi>s</mi><mtext> </mtext><mi>p</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p05">Obtain the amount of the task completed by multiplying the work rate by the amount of time the painter works. Typically, work-rate problems involve people or machines working together to complete tasks. In general, if <em class="emphasis">t</em> represents the time two people work together, then we have the following <span class="margin_term"><a class="glossterm">work-rate formula</a><span class="glossdef"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2456" display="inline"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mo>⋅</mo><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac><mo>⋅</mo><mi>t</mi><mo>=</mo><mn>1</mn></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2457" display="inline"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2458" display="inline"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac></mrow></math></span> are the individual work rates and <em class="emphasis">t</em> is the time it takes to complete the task working together.</span></span>:</p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2459" display="block"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac><mi>t</mi><mo>=</mo><mstyle color="#007fbf"><mtext> </mtext><mi>a</mi><mi>m</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>o</mi><mi>f</mi><mtext> </mtext><mi>t</mi><mi>a</mi><mi>s</mi><mi>k</mi><mtext> </mtext><mi>c</mi><mi>o</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>e</mi><mi>t</mi><mi>e</mi><mi>d</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mi>g</mi><mi>e</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi></mstyle></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p07">Here <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2460" display="inline"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2461" display="inline"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac></mrow></math></span> are the individual work rates.</p>
<div class="callout block" id="fwk-redden-ch04_s08_s02_n01">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch04_s08_s02_p08">Joe can paint a typical room in 2 hours less time than Mark. If Joe and Mark can paint 5 rooms working together in a 12 hour shift, how long does it take each to paint a single room?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p09">Let <em class="emphasis">x</em> represent the time it takes Mark to paint a typical room.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p10">Let <em class="emphasis">x</em> − 2 represent the time it takes Joe to paint a typical room.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p11">Therefore, Mark’s individual work-rate is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2462" display="inline"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span> rooms per hour and Joe’s is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2463" display="inline"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span> rooms per hour. Both men worked for 12 hours. We can organize the data in a chart, just as we did with distance problems.</p>
<div class="informalfigure large">
<img src="section_07/a3ff23e0360253a82e6e56403f17d147.png">
</div>
<p class="para" id="fwk-redden-ch04_s08_s02_p13">Working together, they can paint 5 total rooms in 12 hours. This leads us to the following algebraic setup:</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p14"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2464" display="block"><mrow><mfrac><mrow><mn>12</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>=</mo><mn>5</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s02_p15">Multiply both sides by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2465" display="inline"><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s02_p16"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2466" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo></mrow></mstyle><mrow><mo>(</mo><mrow><mfrac><mrow><mn>12</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo></mrow></mstyle><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo></mrow></mstyle><mfrac><mrow><mn>12</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo></mrow></mstyle><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo></mrow></mstyle><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>12</mn><mi>x</mi><mo>+</mo><mn>12</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>12</mn><mi>x</mi><mo>+</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>24</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>34</mn><mi>x</mi><mo>+</mo><mn>24</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s02_p17">Solve the resulting quadratic equation by factoring.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p18"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2467" display="block"><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mspace width="2em"></mspace><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>34</mn><mi>x</mi><mo>+</mo><mn>24</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>o</mi><mi>r</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>4</mn><mn>5</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s02_p19">We can disregard <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2468" display="inline"><mrow><mfrac><mn>4</mn><mn>5</mn></mfrac></mrow></math></span> because back substituting into <em class="emphasis">x</em> − 2 would yield a negative time to paint a room. Take <em class="emphasis">x</em> = 6 to be the only solution and use it to find the time it takes Joe to paint a typical room.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p20"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2469" display="block"><mrow><mi>x</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>6</mn><mo>−</mo><mn>2</mn><mo>=</mo><mn>4</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s02_p21">Answer: Joe can paint a typical room in 4 hours and Mark can paint a typical room in 6 hours. As a check we can multiply both work rates by 12 hours to see that together they can paint 5 rooms.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p22"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2470" display="block"><mrow><mrow><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>J</mi><mi>o</mi><mi>e</mi></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi></mrow><mrow><mn>4</mn><mtext> </mtext><mi>h</mi><mi>r</mi><mi>s</mi></mrow></mfrac><mo>⋅</mo><mn>12</mn><mtext> </mtext><mi>h</mi><mi>r</mi><mi>s</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi><mi>s</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>M</mi><mi>a</mi><mi>r</mi><mi>k</mi></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>1</mn><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi></mrow><mrow><mn>6</mn><mtext> </mtext><mi>h</mi><mi>r</mi><mi>s</mi></mrow></mfrac><mo>⋅</mo><mn>12</mn><mtext> </mtext><mi>h</mi><mi>r</mi><mi>s</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi><mi>s</mi></mrow></mtd></mtr></mtable><mtext> </mtext></mrow><mo>}</mo></mrow><mtext> </mtext><mi>T</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mn>5</mn><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>m</mi><mi>s</mi><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s08_s02_n02">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch04_s08_s02_p23">It takes Bill twice as long to lay a tile floor by himself as it does Manny. After working together with Bill for 4 hours, Manny was able to complete the job in 2 additional hours. How long would it have taken Manny working alone?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p24">Let <em class="emphasis">x</em> represent the time it takes Manny to lay the floor alone.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p25">Let 2<em class="emphasis">x</em> represent the time it takes Bill to lay the floor alone.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p26">Manny’s work rate is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2471" display="inline"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span> of the floor per hour and Bill’s work rate is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2472" display="inline"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow><mo>.</mo></math></span> Bill worked on the job for 4 hours and Manny worked on the job for 6 hours.</p>
<div class="informalfigure large">
<img src="section_07/b1c1f8aae7af64a964c183caa9351f06.png">
</div>
<p class="para" id="fwk-redden-ch04_s08_s02_p28">This leads us to the following algebraic setup:</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p29"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2473" display="block"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>⋅</mo><mn>6</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>⋅</mo><mn>4</mn><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s02_p30">Solve.</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p31"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2474" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mfrac><mn>6</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>4</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mi>x</mi><mo>⋅</mo></mstyle><mrow><mo>(</mo><mrow><mfrac><mn>6</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>2</mn><mi>x</mi></mfrac></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>x</mi><mo>⋅</mo></mstyle><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mo>+</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s02_p32">Answer: It would have taken Manny 8 hours to complete the floor by himself.</p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s08_s02_p33">Consider the work-rate formula where one task is to be completed.</p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p34"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2475" display="block"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac><mi>t</mi><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s08_s02_p35">Factor out the time <em class="emphasis">t</em> and then divide both sides by <em class="emphasis">t</em>. This will result in equivalent specialized work-rate formulas:</p>
<p class="para block" id="fwk-redden-ch04_s08_s02_p36"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2476" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>t</mi><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mi>t</mi></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s08_s02_p37">In summary, we have the following equivalent work-rate formulas:</p>
<span class="informalequation block"><math xml:id="fwk-redden-ch04_m2477" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="center"><mstyle color="#007fbf"><mtext> </mtext><mi>W</mi><mi>o</mi><mi>r</mi><mi>k</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi><mi>s</mi></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac><mi>t</mi><mo>=</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac><mi>t</mi><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mi>t</mi><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac><mo>=</mo><mn>1</mn><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac></mtd></mtr></mtable></math></span>
<div class="callout block" id="fwk-redden-ch04_s08_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s08_s02_p39"><strong class="emphasis bold">Try this!</strong> Matt can tile a countertop in 2 hours, and his assistant can do the same job in 3 hours. If Matt starts the job and his assistant joins him 1 hour later, then how long will it take to tile the countertop?</p>
<p class="para" id="fwk-redden-ch04_s08_s02_p40">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2478" display="inline"><mrow><mn>1</mn><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow></math></span> hours</p>
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</div>
</div>
</div>
<div class="section" id="fwk-redden-ch04_s08_s03" version="5.0" lang="en">
<h2 class="title editable block">Solving Problems involving Direct, Inverse, and Joint variation</h2>
<p class="para editable block" id="fwk-redden-ch04_s08_s03_p01">Many real-world problems encountered in the sciences involve two types of functional relationships. The first type can be explored using the fact that the distance <em class="emphasis">s</em> in feet an object falls from rest, without regard to air resistance, can be approximated using the following formula:</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p02"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2479" display="block"><mrow><mi>s</mi><mo>=</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p03">Here <em class="emphasis">t</em> represents the time in seconds the object has been falling. For example, after 2 seconds the object will have fallen <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2480" display="inline"><mrow><mi>s</mi><mo>=</mo><mn>16</mn><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>16</mn><mo>⋅</mo><mn>4</mn><mo>=</mo><mn>64</mn></mrow></math></span> feet.</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p04"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Time <em class="emphasis">t</em> in seconds</p></th>
<th align="center"><p class="para">Distance <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2481" display="inline"><mrow><mi>s</mi><mo>=</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></math></span> in feet</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para">0</p></td>
<td align="center"><p class="para">0</p></td>
</tr>
<tr>
<td align="center"><p class="para">1</p></td>
<td align="center"><p class="para">16</p></td>
</tr>
<tr>
<td align="center"><p class="para">2</p></td>
<td align="center"><p class="para">64</p></td>
</tr>
<tr>
<td align="center"><p class="para">3</p></td>
<td align="center"><p class="para">144</p></td>
</tr>
<tr>
<td align="center"><p class="para">4</p></td>
<td align="center"><p class="para">256</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch04_s08_s03_p05">In this example, we can see that the distance varies over time as the product of a constant 16 and the square of the time <em class="emphasis">t</em>. This relationship is described as <span class="margin_term"><a class="glossterm">direct variation</a><span class="glossdef">Describes two quantities <em class="emphasis">x</em> and <em class="emphasis">y</em> that are constant multiples of each other: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2482" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>k</mi><mi>x</mi></mrow><mo>.</mo></math></span></span></span> and 16 is called the <span class="margin_term"><a class="glossterm">constant of variation</a><span class="glossdef">The nonzero multiple <em class="emphasis">k</em>, when quantities vary directly or inversely.</span></span>. Furthermore, if we divide both sides of <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2483" display="inline"><mrow><mi>s</mi><mo>=</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></math></span> by <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2484" display="inline"><mrow><msup><mi>t</mi><mn>2</mn></msup></mrow></math></span> we have</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p06"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2485" display="block"><mrow><mfrac><mi>s</mi><mrow><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>16</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s08_s03_p07">In this form, it is reasonable to say that <em class="emphasis">s</em> is proportional to <em class="emphasis">t</em><sup class="superscript">2</sup>, and 16 is called the <span class="margin_term"><a class="glossterm">constant of proportionality</a><span class="glossdef">Used when referring to the constant of variation.</span></span>. In general, we have</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p08"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Key words</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Translation</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para">“<em class="emphasis">y</em> varies directly as <em class="emphasis">x</em>”</p></td>
<td rowspan="3" align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2486" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>k</mi><mi>x</mi></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">“<em class="emphasis">y</em> is <strong class="emphasis bold">directly proportional</strong> to <em class="emphasis">x</em>”</p></td>
</tr>
<tr>
<td align="center"><p class="para">“<em class="emphasis">y</em> is proportional to <em class="emphasis">x</em>”</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">Used when referring to direct variation.</span></span></p>
<p class="para editable block" id="fwk-redden-ch04_s08_s03_p09">Here <em class="emphasis">k</em> is nonzero and is called the constant of variation or the constant of proportionality. Typically, we will be given information from which we can determine this constant.</p>
<div class="callout block" id="fwk-redden-ch04_s08_s03_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch04_s08_s03_p10">An object’s weight on Earth varies directly to its weight on the Moon. If a man weighs 180 pounds on Earth, then he will weigh 30 pounds on the Moon. Set up an algebraic equation that expresses the weight on Earth in terms of the weight on the Moon and use it to determine the weight of a woman on the Moon if she weighs 120 pounds on Earth.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p11">Let <em class="emphasis">y</em> represent weight on Earth.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p12">Let <em class="emphasis">x</em> represent weight on the Moon.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p13">We are given that the “weight on Earth varies directly to the weight on the Moon.”</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p14"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2487" display="block"><mrow><mi>y</mi><mo>=</mo><mi>k</mi><mi>x</mi></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p15">To find the constant of variation <em class="emphasis">k</em>, use the given information. A 180-lb man on Earth weighs 30 pounds on the Moon, or <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2488" display="inline"><mrow><mi>y</mi><mi> </mi><mo>=</mo><mi> </mi><mn>180</mn></mrow></math></span> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2489" display="inline"><mrow><mi> </mi><mi>x</mi><mi> </mi><mo>=</mo><mi> </mi><mn>30</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p16"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2490" display="block"><mrow><mn>180</mn><mo>=</mo><mi>k</mi><mo>⋅</mo><mn>30</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p17">Solve for <em class="emphasis">k</em>.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p18"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2491" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>180</mn></mrow><mrow><mn>30</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>k</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>k</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p19">Next, set up a formula that models the given information.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p20"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2492" display="block"><mrow><mi>y</mi><mo>=</mo><mn>6</mn><mi>x</mi></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p21">This implies that a person’s weight on Earth is 6 times his weight on the Moon. To answer the question, use the woman’s weight on Earth, <em class="emphasis">y</em> = 120 lbs, and solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p22"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2493" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>120</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn><mi>x</mi></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>120</mn></mrow><mn>6</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>20</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p23">Answer: The woman weighs 20 pounds on the Moon.</p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s08_s03_p24">The second functional relationship can be explored using the formula that relates the intensity of light <em class="emphasis">I</em> to the distance from its source <em class="emphasis">d</em>.</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p25"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2494" display="block"><mrow><mi>I</mi><mo>=</mo><mfrac><mi>k</mi><mrow><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p26">Here <em class="emphasis">k</em> represents some constant. A foot-candle is a measurement of the intensity of light. One foot-candle is defined to be equal to the amount of illumination produced by a standard candle measured one foot away. For example, a 125-Watt fluorescent growing light is advertised to produce 525 foot-candles of illumination. This means that at a distance <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2495" display="inline"><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow></math></span> foot, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2496" display="inline"><mrow><mi>I</mi><mo>=</mo><mn>525</mn></mrow></math></span> foot-candles and we have:</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p27"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2497" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mn>525</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mi>k</mi><mrow><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>525</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>k</mi></mtd></mtr></mtable></math></span></p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p28">Using <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2498" display="inline"><mrow><mi>k</mi><mo>=</mo><mn>525</mn></mrow></math></span> we can construct a formula which gives the light intensity produced by the bulb:</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p29"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2499" display="block"><mrow><mi>I</mi><mo>=</mo><mfrac><mrow><mn>525</mn></mrow><mrow><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s08_s03_p30">Here <em class="emphasis">d</em> represents the distance the growing light is from the plants. In the following chart, we can see that the amount of illumination fades quickly as the distance from the plants increases.</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p31"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">distance <em class="emphasis">t</em> in feet</p></th>
<th align="center">
<p class="para">Light Intensity</p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2500" display="inline"><mrow><mi>I</mi><mo>=</mo><mfrac><mrow><mn>525</mn></mrow><mrow><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
</th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para">1</p></td>
<td align="center"><p class="para">525</p></td>
</tr>
<tr>
<td align="center"><p class="para">2</p></td>
<td align="center"><p class="para">131.25</p></td>
</tr>
<tr>
<td align="center"><p class="para">3</p></td>
<td align="center"><p class="para">58.33</p></td>
</tr>
<tr>
<td align="center"><p class="para">4</p></td>
<td align="center"><p class="para">32.81</p></td>
</tr>
<tr>
<td align="center"><p class="para">5</p></td>
<td align="center"><p class="para">21</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch04_s08_s03_p32">This type of relationship is described as an <span class="margin_term"><a class="glossterm">inverse variation</a><span class="glossdef">Describes two quantities <em class="emphasis">x</em> and <em class="emphasis">y</em>, where one variable is directly proportional to the reciprocal of the other: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2501" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></mrow><mo>.</mo></math></span></span></span>. We say that <em class="emphasis">I</em> is <span class="margin_term"><a class="glossterm">inversely proportional</a><span class="glossdef">Used when referring to inverse variation.</span></span> to the square of the distance <em class="emphasis">d</em>, where 525 is the constant of proportionality. In general, we have</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p33"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Key words</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Translation</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para">“<em class="emphasis">y</em> varies inversely as <em class="emphasis">x</em>”</p></td>
<td rowspan="2" align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2502" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">“<em class="emphasis">y</em> is inversely proportional to <em class="emphasis">x</em>”</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch04_s08_s03_p34">Again, <em class="emphasis">k</em> is nonzero and is called the constant of variation or the constant of proportionality.</p>
<div class="callout block" id="fwk-redden-ch04_s08_s03_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch04_s08_s03_p35">The weight of an object varies inversely as the square of its distance from the center of Earth. If an object weighs 100 pounds on the surface of Earth (approximately 4,000 miles from the center), how much will it weigh at 1,000 miles above Earth’s surface?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p36">Let <em class="emphasis">w</em> represent the weight of the object.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p37">Let <em class="emphasis">d</em> represent the object’s distance from the center of Earth.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p38">Since “<em class="emphasis">w</em> varies inversely as the square of <em class="emphasis">d</em>,” we can write</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p39"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2503" display="block"><mrow><mi>w</mi><mo>=</mo><mfrac><mi>k</mi><mrow><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p40">Use the given information to find <em class="emphasis">k</em>. An object weighs 100 pounds on the surface of Earth, approximately 4,000 miles from the center. In other words, <em class="emphasis">w</em> = 100 when <em class="emphasis">d</em> = 4,000:</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p41"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2504" display="block"><mrow><mn>100</mn><mo>=</mo><mfrac><mi>k</mi><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>4,000</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p42">Solve for <em class="emphasis">k</em>.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p43"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2505" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4,000</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mn>100</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4,000</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mfrac><mi>k</mi><mrow><msup><mrow><mo stretchy="false">(</mo><mn>4,000</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>1,600,000,000</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>k</mi></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>1.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>k</mi></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p44">Therefore, we can model the problem with the following formula:</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p45"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2506" display="block"><mrow><mi>w</mi><mo>=</mo><mfrac><mrow><mn>1.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow><mrow><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p46">To use the formula to find the weight, we need the distance from the center of Earth. Since the object is 1,000 miles above the surface, find the distance from the center of Earth by adding 4,000 miles:</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p47"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2507" display="block"><mrow><mi>d</mi><mo>=</mo><mn>4,000</mn><mo>+</mo><mn>1,000</mn><mo>=</mo><mn>5,000</mn><mtext> </mtext><mtext> </mtext><mtext>miles</mtext></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p48">To answer the question, use the formula with <em class="emphasis">d</em> = 5,000.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p49"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2508" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>1.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mn>5,000</mn></mrow></mstyle><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>1.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow><mrow><mn>25,000,000</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>1.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup></mrow><mrow><mn>2.5</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>7</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0.64</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>64</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p50">Answer: The object will weigh 64 pounds at a distance 1,000 miles above the surface of Earth.</p>
</div>
<p class="para block" id="fwk-redden-ch04_s08_s03_p51">Lastly, we define relationships between multiple variables, described as <span class="margin_term"><a class="glossterm">joint variation</a><span class="glossdef">Describes a quantity <em class="emphasis">y</em> that varies directly as the product of two other quantities <em class="emphasis">x</em> and <em class="emphasis">z</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2509" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>k</mi><mi>x</mi><mi>z</mi></mrow><mo>.</mo></math></span></span></span>. In general, we have</p>
<p class="para block" id="fwk-redden-ch04_s08_s03_p52"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Key Words</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Translation</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para">“<em class="emphasis">y</em> varies jointly as <em class="emphasis">x</em> and <em class="emphasis">z</em>”</p></td>
<td rowspan="2" align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2510" display="inline"><mrow><mi>y</mi><mo>=</mo><mi>k</mi><mi>x</mi><mi>z</mi></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">“<em class="emphasis">y</em> is <strong class="emphasis bold">jointly proportional</strong> to <em class="emphasis">x</em> and <em class="emphasis">z</em>”</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">Used when referring to joint variation.</span></span></p>
<p class="para editable block" id="fwk-redden-ch04_s08_s03_p53">Here <em class="emphasis">k</em> is nonzero and is called the constant of variation or the constant of proportionality.</p>
<div class="callout block" id="fwk-redden-ch04_s08_s03_n03">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch04_s08_s03_p54">The area of an ellipse varies jointly as <em class="emphasis">a</em>, half of the ellipse’s major axis, and <em class="emphasis">b</em>, half of the ellipse’s minor axis as pictured. If the area of an ellipse is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2511" display="inline"><mrow><mn>300</mn><mi>π</mi><mtext> </mtext><msup><mrow><mtext>cm</mtext></mrow><mn>2</mn></msup></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2512" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>10</mn><mtext> </mtext><mtext>cm</mtext></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2513" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>30</mn><mtext> </mtext><mtext>cm</mtext></mrow></math></span>, what is the constant of proportionality? Give a formula for the area of an ellipse.</p>
<div class="informalfigure large">
<img src="section_07/06f2832ec18c151800b5823d1adebb25.png">
</div>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p56">If we let <em class="emphasis">A</em> represent the area of an ellipse, then we can use the statement “area varies jointly as <em class="emphasis">a</em> and <em class="emphasis">b</em>” to write</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p57"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2514" display="block"><mrow><mi>A</mi><mo>=</mo><mi>k</mi><mi>a</mi><mi>b</mi></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p58">To find the constant of variation <em class="emphasis">k</em>, use the fact that the area is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2515" display="inline"><mrow><mn>300</mn><mi>π</mi></mrow></math></span> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2516" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>10</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2517" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>30</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p59"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2518" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>300</mn><mi>π</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>k</mi><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mn>10</mn></mrow></mstyle><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mstyle color="#007f3f"><mrow><mn>30</mn></mrow></mstyle><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>300</mn><mi>π</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>300</mn><mi>k</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>π</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>k</mi></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p60">Therefore, the formula for the area of an ellipse is</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p61"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2519" display="block"><mrow><mi>A</mi><mo>=</mo><mi>π</mi><mi>a</mi><mi>b</mi></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s08_s03_p62">Answer: The constant of proportionality is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2520" display="inline"><mi>π</mi></math></span> and the formula for the area of an ellipse is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2521" display="inline"><mrow><mi>A</mi><mo>=</mo><mi>a</mi><mi>b</mi><mi>π</mi></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s08_s03_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s08_s03_p63"><strong class="emphasis bold">Try this!</strong> Given that <em class="emphasis">y</em> varies directly as the square of <em class="emphasis">x</em> and inversely with <em class="emphasis">z</em>, where <em class="emphasis">y</em> = 2 when <em class="emphasis">x</em> = 3 and <em class="emphasis">z</em> = 27, find <em class="emphasis">y</em> when <em class="emphasis">x</em> = 2 and <em class="emphasis">z</em> = 16.</p>
<p class="para" id="fwk-redden-ch04_s08_s03_p64">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2522" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/ee3AFf7b6Kg" condition="http://img.youtube.com/vi/ee3AFf7b6Kg/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/ee3AFf7b6Kg" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch04_s08_s03_n04">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch04_s08_s03_l01" mark="bullet">
<li>When solving distance problems where the time element is unknown, use the equivalent form of the uniform motion formula, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2523" display="inline"><mrow><mi>t</mi><mo>=</mo><mfrac><mi>D</mi><mi>r</mi></mfrac></mrow></math></span>, to avoid introducing more variables.</li>
<li>When solving work-rate problems, multiply the individual work rate by the time to obtain the portion of the task completed. The sum of the portions of the task results in the total amount of work completed.</li>
<li>The setup of variation problems usually requires multiple steps. First, identify the key words to set up an equation and then use the given information to find the constant of variation <em class="emphasis">k</em>. After determining the constant of variation, write a formula that models the problem. Once a formula is found, use it to answer the question.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s08_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd01">
<h3 class="title">Part A: Solving Uniform Motion Problems</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch04_s08_qs01_p01"><strong class="emphasis bold">Use algebra to solve the following applications.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p02">Every morning Jim spends 1 hour exercising. He runs 2 miles and then he bikes 16 miles. If Jim can bike twice as fast as he can run, at what speed does he average on his bike?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p04">Sally runs 3 times as fast as she walks. She ran for <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2524" display="inline"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span> of a mile and then walked another <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2525" display="inline"><mrow><mn>3</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> miles. The total workout took <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2526" display="inline"><mrow><mn>1</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> hours. What was Sally’s average walking speed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p06">On a business trip, an executive traveled 720 miles by jet and then another 80 miles by helicopter. If the jet averaged 3 times the speed of the helicopter, and the total trip took 4 hours, what was the average speed of the jet?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p08">A triathlete can run 3 times as fast as she can swim and bike 6 times as fast as she can swim. The race consists of a <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2528" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span> mile swim, 3 mile run, and a 12 mile bike race. If she can complete all of these events in <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2529" display="inline"><mrow><mn>1</mn><mfrac><mn>5</mn><mn>8</mn></mfrac></mrow></math></span> hour, then how fast can she swim, run and bike?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p10">On a road trip, Marty was able to drive an average 4 miles per hour faster than George. If Marty was able to drive 39 miles in the same amount of time George drove 36 miles, what was Marty’s average speed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p12">The bus is 8 miles per hour faster than the trolley. If the bus travels 9 miles in the same amount of time the trolley can travel 7 miles, what is the average speed of each?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p14">Terry decided to jog the 5 miles to town. On the return trip, she walked the 5 miles home at half of the speed that she was able to jog. If the total trip took 3 hours, what was her average jogging speed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p16">James drove the 24 miles to town and back in 1 hour. On the return trip, he was able to average 20 miles per hour faster than he averaged on the trip to town. What was his average speed on the trip to town?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p18">A light aircraft was able to travel 189 miles with a 14 mile per hour tailwind in the same time it was able to travel 147 miles against it. What was the speed of the aircraft in calm air?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p20">A jet flew 875 miles with a 30 mile per hour tailwind. On the return trip, against a 30 mile per hour headwind, it was able to cover only 725 miles in the same amount of time. How fast was the jet in calm air?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p22">A helicopter averaged 90 miles per hour in calm air. Flying with the wind it was able to travel 250 miles in the same amount of time it took to travel 200 miles against it. What is the speed of the wind?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p24">Mary and Joe took a road-trip on separate motorcycles. Mary’s average speed was 12 miles per hour less than Joe’s average speed. If Mary drove 115 miles in the same time it took Joe to drive 145 miles, what was Mary’s average speed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p26">A boat averaged 12 miles per hour in still water. On a trip downstream, with the current, the boat was able to travel 26 miles. The boat then turned around and returned upstream 33 miles. How fast was the current if the total trip took 5 hours?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p28">If the river current flows at an average 3 miles per hour, a tour boat can make an 18-mile tour downstream with the current and back the 18 miles against the current in <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2530" display="inline"><mrow><mn>4</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> hours. What is the average speed of the boat in still water?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p30">Jose drove 10 miles to his grandmother’s house for dinner and back that same evening. Because of traffic, he averaged 20 miles per hour less on the return trip. If it took <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2531" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span> hour longer to get home, what was his average speed driving to his grandmother’s house?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p32">Jerry paddled his kayak, upstream against a 1 mph current, for 12 miles. The return trip, downstream with the 1 mph current, took one hour less time. How fast did Jerry paddle the kayak in still water?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p34">James and Mildred left the same location in separate cars and met in Los Angeles 300 miles away. James was able to average 10 miles an hour faster than Mildred on the trip. If James arrived 1 hour earlier than Mildred, what was Mildred’s average speed?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p36">A bus is 20 miles per hour faster than a bicycle. If Bill boards a bus at the same time and place that Mary departs on her bicycle, Bill will arrive downtown 5 miles away <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2532" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span> hour earlier than Mary. What is the average speed of the bus?</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd02">
<h3 class="title">Part B: Solving Work-Rate Problems</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd02_qd01" start="19">
<p class="para" id="fwk-redden-ch04_s08_qs01_p38"><strong class="emphasis bold">Use algebra to solve the following applications.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p39">Mike can paint the office by himself in <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2533" display="inline"><mrow><mn>4</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> hours. Jordan can paint the office in 6 hours. How long will it take them to paint the office working together?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p41">Barry can lay a brick driveway by himself in <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2535" display="inline"><mrow><mn>3</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> days. Robert does the same job in 5 days. How long will it take them to lay the brick driveway working together?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p43">A larger pipe fills a water tank twice as fast as a smaller pipe. When both pipes are used, they fill the tank in 10 hours. If the larger pipe is left off, how long would it take the smaller pipe to fill the tank?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p45">A newer printer can print twice as fast as an older printer. If both printers working together can print a batch of flyers in 45 minutes, then how long would it take the older printer to print the batch working alone?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p47">Mary can assemble a bicycle for display in 2 hours. It takes Jane 3 hours to assemble a bicycle. How long will it take Mary and Jane, working together, to assemble 5 bicycles?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p49">Working alone, James takes twice as long to assemble a computer as it takes Bill. In one 8-hour shift, working together, James and Bill can assemble 6 computers. How long would it take James to assemble a computer if he were working alone?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p51">Working alone, it takes Harry one hour longer than Mike to install a fountain. Together they can install 10 fountains in 12 hours. How long would it take Mike to install 10 fountains by himself?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p53">Working alone, it takes Henry 2 hours longer than Bill to paint a room. Working together they painted <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2537" display="inline"><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span> rooms in 6 hours. How long would it have taken Henry to paint the same amount if he were working alone?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p55">Manny, working alone, can install a custom cabinet in 3 hours less time than his assistant. Working together they can install the cabinet in 2 hours. How long would it take Manny to install the cabinet working alone?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p57">Working alone, Garret can assemble a garden shed in 5 hours less time than his brother. Working together, they need 6 hours to build the garden shed. How long would it take Garret to build the shed working alone?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p59">Working alone, the assistant-manager takes 2 more hours than the manager to record the inventory of the entire shop. After working together for 2 hours, it took the assistant-manager 1 additional hour to complete the inventory. How long would it have taken the manager to complete the inventory working alone?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p61">An older printer can print a batch of sales brochures in 16 minutes. A newer printer can print the same batch in 10 minutes. After working together for some time, the newer printer was shut down and it took the older printer 3 more minutes to complete the job. How long was the newer printer operating?</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03">
<h3 class="title">Part C: Solving Variation Problems</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03_qd01" start="31">
<p class="para" id="fwk-redden-ch04_s08_qs01_p63"><strong class="emphasis bold">Translate each of the following sentences into a mathematical formula.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p64">The distance <em class="emphasis">D</em> an automobile can travel is directly proportional to the time <em class="emphasis">t</em> that it travels at a constant speed.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p66">The extension of a hanging spring <em class="emphasis">d</em> is directly proportional to the weight <em class="emphasis">w</em> attached to it.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p68">An automobile’s braking distance <em class="emphasis">d</em> is directly proportional to the square of the automobile’s speed <em class="emphasis">v</em>.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p70">The volume <em class="emphasis">V</em> of a sphere varies directly as the cube of its radius <em class="emphasis">r</em>.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p72">The volume <em class="emphasis">V</em> of a given mass of gas is inversely proportional to the pressure <em class="emphasis">p</em> exerted on it.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p74">Every particle of matter in the universe attracts every other particle with a force <em class="emphasis">F</em> that is directly proportional to the product of the masses <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2543" display="inline"><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2544" display="inline"><mrow><msub><mi>m</mi><mn>2</mn></msub></mrow></math></span> of the particles, and it is inversely proportional to the square of the distance <em class="emphasis">d</em> between them.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p76">Simple interest <em class="emphasis">I</em> is jointly proportional to the annual interest rate <em class="emphasis">r</em> and the time <em class="emphasis">t</em> in years a fixed amount of money is invested.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p78">The time <em class="emphasis">t</em> it takes an object to fall is directly proportional to the square root of the distance <em class="emphasis">d</em> it falls.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03_qd02" start="39">
<p class="para" id="fwk-redden-ch04_s08_qs01_p80"><strong class="emphasis bold">Construct a mathematical model given the following:</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p81"><em class="emphasis">y</em> varies directly as <em class="emphasis">x</em>, and <em class="emphasis">y</em> = 30 when <em class="emphasis">x</em> = 6.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p83"><em class="emphasis">y</em> varies directly as <em class="emphasis">x</em>, and <em class="emphasis">y</em> = 52 when <em class="emphasis">x</em> = 4.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p85"><em class="emphasis">y</em> is directly proportional to <em class="emphasis">x</em>, and <em class="emphasis">y</em> = 12 when <em class="emphasis">x</em> = 3.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p87"><em class="emphasis">y</em> is directly proportional to <em class="emphasis">x</em>, and <em class="emphasis">y</em> = 120 when <em class="emphasis">x</em> = 20.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p89"><em class="emphasis">y</em> is inversely proportional to <em class="emphasis">x</em>, and <em class="emphasis">y</em> = 3 when <em class="emphasis">x</em> = 9.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p91"><em class="emphasis">y</em> is inversely proportional to <em class="emphasis">x</em>, and <em class="emphasis">y</em> = 21 when <em class="emphasis">x</em> = 3.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p93"><em class="emphasis">y</em> varies inversely as <em class="emphasis">x</em>, and <em class="emphasis">y</em> = 2 when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2554" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mn>8</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p95"><em class="emphasis">y</em> varies inversely as <em class="emphasis">x</em>, and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2556" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mn>2</mn></mfrac></mrow></math></span> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2557" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mn>9</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p97"><em class="emphasis">y</em> is jointly proportional to <em class="emphasis">x</em> and <em class="emphasis">z</em>, where <em class="emphasis">y</em> = 2 when <em class="emphasis">x</em> = 1 and <em class="emphasis">z</em> = 3.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p99"><em class="emphasis">y</em> is jointly proportional to <em class="emphasis">x</em> and <em class="emphasis">z</em>, where <em class="emphasis">y</em> = 15 when <em class="emphasis">x</em> = 3 and <em class="emphasis">z</em> = 7.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p101"><em class="emphasis">y</em> varies jointly as <em class="emphasis">x</em> and <em class="emphasis">z</em>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2561" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mn>3</mn></mfrac></mrow></math></span> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2562" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></math></span> and <em class="emphasis">z</em> = 12.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p103"><em class="emphasis">y</em> varies jointly as <em class="emphasis">x</em> and <em class="emphasis">z</em>, where <em class="emphasis">y</em> = 5 when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2564" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mn>2</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2565" display="inline"><mrow><mi>z</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mn>9</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p105"><em class="emphasis">y</em> varies directly as the square of <em class="emphasis">x</em>, where <em class="emphasis">y</em> = 45 when <em class="emphasis">x</em> = 3.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p107"><em class="emphasis">y</em> varies directly as the square of <em class="emphasis">x</em>, where <em class="emphasis">y</em> = 3 when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2568" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p109"><em class="emphasis">y</em> is inversely proportional to the square of <em class="emphasis">x</em>, where <em class="emphasis">y</em> = 27 when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2570" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p111"><em class="emphasis">y</em> is inversely proportional to the square of <em class="emphasis">x</em>, where <em class="emphasis">y</em> = 9 when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2572" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p113"><em class="emphasis">y</em> varies jointly as <em class="emphasis">x</em> and the square of <em class="emphasis">z</em>, where <em class="emphasis">y</em> = 6 when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2574" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mn>4</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2575" display="inline"><mrow><mi>z</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p115"><em class="emphasis">y</em> varies jointly as <em class="emphasis">x</em> and <em class="emphasis">z</em> and inversely as the square of <em class="emphasis">w</em>, where <em class="emphasis">y</em> = 5 when <em class="emphasis">x</em> = 1, <em class="emphasis">z</em> = 3, and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2577" display="inline"><mrow><mi>w</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p117"><em class="emphasis">y</em> varies directly as the square root of <em class="emphasis">x</em> and inversely as the square of <em class="emphasis">z</em>, where <em class="emphasis">y</em> = 15 when <em class="emphasis">x</em> = 25 and <em class="emphasis">z</em> = 2.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p119"><em class="emphasis">y</em> varies directly as the square of <em class="emphasis">x</em> and inversely as <em class="emphasis">z</em> and the square of <em class="emphasis">w</em>, where <em class="emphasis">y</em> = 14 when <em class="emphasis">x</em> = 4, <em class="emphasis">w</em> = 2, and <em class="emphasis">z</em> = 2.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03_qd03" start="59">
<p class="para" id="fwk-redden-ch04_s08_qs01_p121"><strong class="emphasis bold">Solve applications involving variation.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p122">Revenue in dollars is directly proportional to the number of branded sweatshirts sold. The revenue earned from selling 25 sweatshirts is $318.75. Determine the revenue if 30 sweatshirts are sold.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p124">The sales tax on the purchase of a new car varies directly as the price of the car. If an $18,000 new car is purchased, then the sales tax is $1,350. How much sales tax is charged if the new car is priced at $22,000?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p126">The price of a share of common stock in a company is directly proportional to the earnings per share (EPS) of the previous 12 months. If the price of a share of common stock in a company is $22.55, and the EPS is published to be $1.10, determine the value of the stock if the EPS increases by $0.20.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p128">The distance traveled on a road trip varies directly with the time spent on the road. If a 126-mile trip can be made in 3 hours, then what distance can be traveled in 4 hours?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p130">The circumference of a circle is directly proportional to its radius. The circumference of a circle with radius 7 centimeters is measured as <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2581" display="inline"><mrow><mn>14</mn><mi>π</mi></mrow></math></span> centimeters. What is the constant of proportionality?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p132">The area of circle varies directly as the square of its radius. The area of a circle with radius 7 centimeters is determined to be <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2583" display="inline"><mrow><mn>49</mn><mi>π</mi></mrow></math></span> square centimeters. What is the constant of proportionality?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p134">The surface area of a sphere varies directly as the square of its radius. When the radius of a sphere measures 2 meters, the surface area measures <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2585" display="inline"><mrow><mn>16</mn><mi>π</mi></mrow></math></span> square meters. Find the surface area of a sphere with radius 3 meters.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p136">The volume of a sphere varies directly as the cube of its radius. When the radius of a sphere measures 3 meters, the volume is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2587" display="inline"><mrow><mn>36</mn><mi>π</mi></mrow></math></span> cubic meters. Find the volume of a sphere with radius 1 meter.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p138">With a fixed height, the volume of a cone is directly proportional to the square of the radius at the base. When the radius at the base measures 10 centimeters, the volume is 200 cubic centimeters. Determine the volume of the cone if the radius of the base is halved.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p140">The distance <em class="emphasis">d</em> an object in free fall drops varies directly with the square of the time <em class="emphasis">t</em> that it has been falling. If an object in free fall drops 36 feet in 1.5 seconds, then how far will it have fallen in 3 seconds?</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03_qd04" start="69">
<p class="para" id="fwk-redden-ch04_s08_qs01_p142"><strong class="emphasis bold">Hooke’s law suggests that the extension of a hanging spring is directly proportional to the weight attached to it. The constant of variation is called the spring constant.</strong></p>
<div class="figure medium" id="fwk-redden-ch04_s08_f01">
<p class="title"><span class="title-prefix">Figure 4.1</span> </p>
<img src="section_07/67a5628e2746f71bce1f5c9fa22d18ba.png">
<p class="para">Robert Hooke (1635—1703)</p>
</div>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p144">A hanging spring is stretched 5 inches when a 20-pound weight is attached to it. Determine its spring constant.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p146">A hanging spring is stretched 3 centimeters when a 2-kilogram weight is attached to it. Determine the spring constant.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p148">If a hanging spring is stretched 3 inches when a 2-pound weight is attached, how far will it stretch with a 5-pound weight attached?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p150">If a hanging spring is stretched 6 centimeters when a 4-kilogram weight is attached to it, how far will it stretch with a 2-kilogram weight attached?</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03_qd05" start="73">
<p class="para" id="fwk-redden-ch04_s08_qs01_p152"><strong class="emphasis bold">The braking distance of an automobile is directly proportional to the square of its speed.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p153">It takes 36 feet to stop a particular automobile moving at a speed of 30 miles per hour. How much breaking distance is required if the speed is 35 miles per hour?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p155">After an accident, it was determined that it took a driver 80 feet to stop his car. In an experiment under similar conditions, it takes 45 feet to stop the car moving at a speed of 30 miles per hour. Estimate how fast the driver was moving before the accident.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03_qd06" start="75">
<div class="figure medium" id="fwk-redden-ch04_s08_f02">
<p class="title"><span class="title-prefix">Figure 4.2</span> </p>
<img src="section_07/a24bffc969070eded5452b0777e9a1ab.png">
<p class="para"> </p>
<p class="para">Robert Boyle (1627—1691)</p>
</div>
<p class="para" id="fwk-redden-ch04_s08_qs01_p157"><strong class="emphasis bold">Boyle’s law states that if the temperature remains constant, the volume <em class="emphasis">V</em> of a given mass of gas is inversely proportional to the pressure <em class="emphasis">p</em> exerted on it.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p159">A balloon is filled to a volume of 216 cubic inches on a diving boat under 1 atmosphere of pressure. If the balloon is taken underwater approximately 33 feet, where the pressure measures 2 atmospheres, then what is the volume of the balloon?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p161">A balloon is filled to 216 cubic inches under a pressure of 3 atmospheres at a depth of 66 feet. What would the volume be at the surface, where the pressure is 1 atmosphere?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p163">To balance a seesaw, the distance from the fulcrum that a person must sit is inversely proportional to his weight. If a 72-pound boy is sitting 3 feet from the fulcrum, how far from the fulcrum must a 54-pound boy sit to balance the seesaw?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p165">The current <em class="emphasis">I</em> in an electrical conductor is inversely proportional to its resistance <em class="emphasis">R</em>. If the current is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2591" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span> ampere when the resistance is 100 ohms, what is the current when the resistance is 150 ohms?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p167">The amount of illumination <em class="emphasis">I</em> is inversely proportional to the square of the distance <em class="emphasis">d</em> from a light source. If 70 foot-candles of illumination is measured 2 feet away from a lamp, what level of illumination might we expect <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2593" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> foot away from the lamp?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p169">The amount of illumination <em class="emphasis">I</em> is inversely proportional to the square of the distance <em class="emphasis">d</em> from a light source. If 40 foot-candles of illumination is measured 3 feet away from a lamp, at what distance can we expect 10 foot-candles of illumination?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p171">The number of men, represented by <em class="emphasis">y</em>, needed to lay a cobblestone driveway is directly proportional to the area <em class="emphasis">A</em> of the driveway and inversely proportional to the amount of time <em class="emphasis">t</em> allowed to complete the job. Typically, 3 men can lay 1,200 square feet of cobblestone in 4 hours. How many men will be required to lay 2,400 square feet of cobblestone in 6 hours?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p173">The volume of a right circular cylinder varies jointly as the square of its radius and its height. A right circular cylinder with a 3-centimeter radius and a height of 4 centimeters has a volume of <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2594" display="inline"><mrow><mn>36</mn><mi>π</mi></mrow></math></span> cubic centimeters. Find a formula for the volume of a right circular cylinder in terms of its radius and height.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p175">The period <em class="emphasis">T</em> of a pendulum is directly proportional to the square root of its length <em class="emphasis">L</em>. If the length of a pendulum is 1 meter, then the period is approximately 2 seconds. Approximate the period of a pendulum that is 0.5 meter in length.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p177">The time <em class="emphasis">t</em> it takes an object to fall is directly proportional to the square root of the distance <em class="emphasis">d</em> it falls. An object dropped from 4 feet will take <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2596" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> second to hit the ground. How long will it take an object dropped from 16 feet to hit the ground?</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s08_qs01_qd03_qd07" start="85">
<p class="para" id="fwk-redden-ch04_s08_qs01_p179"><strong class="emphasis bold">Newton’s universal law of gravitation states that every particle of matter in the universe attracts every other particle with a force <em class="emphasis">F</em> that is directly proportional to the product of the masses</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2597" display="inline"><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow></math></span> <strong class="emphasis bold">and</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2598" display="inline"><mrow><msub><mi>m</mi><mn>2</mn></msub></mrow></math></span> <strong class="emphasis bold">of the particles and inversely proportional to the square of the distance <em class="emphasis">d</em> between them. The constant of proportionality is called the gravitational constant.</strong></p>
<div class="figure medium" id="fwk-redden-ch04_s08_f03">
<p class="title"><span class="title-prefix">Figure 4.3</span> </p>
<img src="section_07/b0f8aa54b2a68270687029a67a7d1e6a.png">
<p class="para"> </p>
<p class="para">Sir Isaac Newton (1643—1727)</p>
<p class="para"> </p>
<p class="para">Source: Portrait of Isaac Newton by Sir Godfrey Kneller, from http://commons.wikimedia.org/wiki/File:GodfreyKneller-IsaacNewton-1689.</p>
<p class="para">http://commons.wikimedia.org/wiki/File:Frans_Hals_-_Portret_</p>
<p class="para">_van_Ren%C3%A9_Descartes.jpg.</p>
</div>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p181">If two objects with masses 50 kilograms and 100 kilograms are <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2599" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> meter apart, then they produce approximately <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2600" display="inline"><mrow><mn>1.34</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></math></span> newtons (N) of force. Calculate the gravitational constant.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p183">Use the gravitational constant from the previous exercise to write a formula that approximates the force <em class="emphasis">F</em> in newtons between two masses <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2602" display="inline"><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2603" display="inline"><mrow><msub><mi>m</mi><mn>2</mn></msub></mrow></math></span>, expressed in kilograms, given the distance <em class="emphasis">d</em> between them in meters.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p185">Calculate the force in newtons between Earth and the Moon, given that the mass of the Moon is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2605" display="inline"><mrow><mn>7.3</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>22</mn></mrow></msup></mrow></math></span> kilograms, the mass of Earth is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2606" display="inline"><mrow><mn>6.0</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></mrow></math></span> kilograms, and the distance between them is on average <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2607" display="inline"><mrow><mn>1.5</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>11</mn></mrow></msup></mrow></math></span> meters.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p187">Calculate the force in newtons between Earth and the Sun, given that the mass of the Sun is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2609" display="inline"><mrow><mn>2.0</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>30</mn></mrow></msup></mrow></math></span> kilograms, the mass of Earth is approximately <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2610" display="inline"><mrow><mn>6.0</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></mrow></math></span> kilograms, and the distance between them is on average <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2611" display="inline"><mrow><mn>3.85</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>8</mn></msup></mrow></math></span> meters.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p189">If <em class="emphasis">y</em> varies directly as the square of <em class="emphasis">x</em>, then how does <em class="emphasis">y</em> change if <em class="emphasis">x</em> is doubled?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p191">If <em class="emphasis">y</em> varies inversely as square of <em class="emphasis">t</em>, then how does <em class="emphasis">y</em> change if <em class="emphasis">t</em> is doubled?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch04_s08_qs01_p193">If <em class="emphasis">y</em> varies directly as the square of <em class="emphasis">x</em> and inversely as the square of <em class="emphasis">t</em>, then how does <em class="emphasis">y</em> change if both <em class="emphasis">x</em> and <em class="emphasis">t</em> are doubled?</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s08_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p03_ans">20 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p07_ans">240 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p11_ans">52 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p15_ans">5 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p19_ans">112 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p23_ans">10 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p27_ans">1 mile per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p31_ans">40 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p35_ans">50 miles per hour</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="19">
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p40_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2534" display="inline"><mrow><mn>2</mn><mfrac><mn>4</mn><mn>7</mn></mfrac></mrow></math></span> hours</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa21_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s08_qs01_p44_ans">30 hours</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s08_qs01_qa23_ans">