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250 lines
9.6 KiB
HTML
250 lines
9.6 KiB
HTML
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<head>
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<title>Activity: Soup Can</title>
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<script>reSpell=[["liter","litre"],["milliliters","millilitres"],["centimeters","centimetres"]];</script>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity:
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Soup Can</h1>
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<p class="center"><img alt="soup can" src="images/soup-can-1.svg" /></p>
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<p>"Pi Soups Incorporated" want to sell soup in cylindrical
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cans that contain <b>400 ml</b> (milliliters)</p>
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<p>They also want to <b>minimize</b> the amount of metal in each can.</p>
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<p>As the local mathematics expert it is your job to help them!</p>
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<h2>Volume and Surface Area</h2>
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<table border="0">
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<tr>
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<td><img alt="cylinder r and h" src="images/soup-can-2.svg" /></td>
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<td width="30"> </td>
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<td><p>For
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a <a href="../geometry/cylinder.html">cylinder</a> of height <span class="large">h</span> and base
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radius <span class="large">r</span>:</p>
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<ul>
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<li>Surface
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Area = 2 × <span class="times">π</span> × r ×
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(r + h)</li>
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<li>Volume = <span class="times">π</span> × r<sup>2 </sup>×
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h</li>
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</ul>
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<p>Which
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can be abbreviated to:</p>
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<ul>
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<li>A = 2<span class="times">π</span>r(r + h)</li>
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<li>V = <span class="times">π</span>r<sup>2</sup>h</li>
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</ul></td>
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</tr>
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</table>
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<h2>Thickness of Metal</h2>
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<p>Let us assume that the metal has uniform thickness, so we just want to find</p>
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<div class="center80">
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<p>The smallest surface area that will hold 400 ml</p>
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</div>
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<p>Let us also assume that the thickness of the metal does not affect the volume (i.e. it is very small in comparison to the size of the can.)</p>
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<p><br style="font-weight: normal;" />
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To
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solve this we will work
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in centimeters, so:</p>
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<ul>
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<li>the area is measured in square centimeters (cm<sup
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style="font-weight: normal;">2</sup>)
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and</li>
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<li>the volume is measured in cubic centimeters (cm<sup
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style="font-weight: normal;">3</sup>).<br
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style="font-weight: normal;" /></li>
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</ul>
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<p>This makes it easy, as <b>1 ml</b> is
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the same as a volume of <b>1 cm<sup style="font-weight: normal;">3</sup></b> - see
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the page <a style="font-weight: normal;"
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href="../measure/metric-volume.html">Metric
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Volume</a>.</p>
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<div class="center80">
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<p>So
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V = 400</p>
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</div>
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<p>This gives us the formula 400 = <span class="times">π</span>r<sup>2</sup>h</p>
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<p>Make h <a
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href="../algebra/equation-formula.html">subject
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of the formula</a> :</p>
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<div class="so">Divide both
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sides by <span class="times">π</span>r<sup style="font-weight: normal;">2</sup>: 400/<span class="times">π</span>r<sup
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style="font-weight: bold;">2</sup> = h</div>
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<div class="so">Swap Sides: h = 400/<span class="times">π</span>r<sup
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style="font-weight: bold;">2</sup></div>
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<p><a href="../algebra/substitution.html">Substitute</a> this value for <b>h</b> into the formula for A:</p>
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<div class="so"> A = 2<i><span class="times">π</span></i>r(r+h) = 2<i><span class="times">π</span></i>r(r + 400/<i><span class="times">π</span></i>r<sup
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style="font-weight: normal;">2</sup>) = 2<i><span class="times">π</span></i>r<sup
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style="font-weight: normal;">2</sup> + 800/r</div>
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<p>Now we know the Area in terms of the Radius:</p>
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<div class="center80">
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<p>A = 2<i><span class="times">π</span></i>r<sup
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style="font-weight: normal;">2</sup> + 800/r</p>
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</div>
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<p>This is our very simple <a href="../algebra/mathematical-models.html">Mathematical Model</a> of the soup can.</p>
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<h2>Your Turn Now!</h2>
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<p>I have helped you enough. Now it is your turn.</p>
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<p>You can use the formula above to investigate how changing the radius affects the area, and to find which radius gives the least area of metal.</p>
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<p>You can make a report that explains things to everyone (pretend you will have to present this to a meeting of executives), with a graph and a table, and words.</p>
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<div class="dotpoint">
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<p>Graph: draw the graph of the function, find the <b>minimum value</b> (you can use the <a href="../data/function-grapher9cee.html?func1=2pi(x)^2+800/x&xmin=-3.9&xmax=13.5&ymin=-46&ymax=841">Function Grapher</a>)</p>
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</div>
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<div class="dotpoint">
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<p>Table: find example points on the graph by <a
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href="../algebra/substitution.html">substituting</a> different values
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of r.</p>
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<div class="example">
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<p>For example, if r = 2, then A = 2<i><span class="times">π</span> ×</i> 2<sup style="font-weight: normal;">2</sup> + 800/2 = 8<i><span class="times">π</span> </i>+
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400 = 425 to the
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nearest whole number.</p>
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<p>In other words, when the
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radius of the can is 2cm, the area of
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metal needed to make the can is 425cm<sup style="font-weight: normal;">2</sup>.</p>
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</div>
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<p>Put the example values into a table, like this one:</p>
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<table border="1">
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<tr>
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<td width="100" height="30" align="center">Radius</td>
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<td width="100" align="center">Area</td>
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<td width="300">Comments</td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">0</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">1</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">2</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">3</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">4</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">5</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">6</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td height="30" align="center"><span class="large">7</span></td>
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<td> </td>
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<td> </td>
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</tr>
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</table>
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</div>
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<div class="dotpoint">
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<p>Report: Put the graph and table together into a report, with:</p>
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<ul>
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<li>an introduction,</li>
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<li>an explanation</li>
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<li>and your conclusions and recommendations.</li>
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</ul>
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</div>
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<h2>A 1 Liter Can</h2>
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<p>What is the best radius for a 1 Liter Can of soup?</p>
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<p> </p>
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<h2>Improving The Model</h2>
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<h3>Air Gap</h3>
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<p>Soup cans are not completely filled with soup ... there is a small air gap on top. You can model this by adding (say) 1cm to the height.</p>
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<p>Can you write the formula including this change?</p>
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<p>How does this affect the graph, and the minimum value?</p>
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<h3>Rims</h3>
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<p>Soup cans have extra thick rims at top and bottom, how could you include this in the formula?</p>
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<p> </p>
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<div class="related">
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<a href="../geometry/cylinder.html">Cylinder</a>
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<a href="index.html">Activity Index</a>
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Copyright © 2020 MathsIsFun.com
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