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324 lines
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HTML
324 lines
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<title>Mathematical Models</title>
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<article id="content" role="main">
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<h1 class="center">Mathematical Models</h1>
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<p>Mathematics can be used to "model", or represent, how the real world works.</p>
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<div class="example">
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<h3>Example: how much space is inside this cardboard box?</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cardboard-box.svg" alt="Cardboard Box" height="200" width="291"></p>
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<p> </p>
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<p>We know three measurements:</p>
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<ul>
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<li><b>l</b> (length),</li>
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<li><b>w</b> (width), and</li>
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<li><b>h</b> (height),</li>
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</ul>
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<p>and the formula for the <a href="../cuboid.html">volume of a cuboid</a> is:</p>
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<p class="center"><span class="larger">Volume = l × w × h</span></p>
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</div>
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<p>So we have a (very simple) mathematical model of the space in that box.</p>
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<h2>Accurate?</h2>
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<p class="larger">The model is not the same as the real thing.</p>
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<p>In our example we did not think about the thickness of the cardboard, or many other "real world" things.</p>
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<p class="center larger">But hopefully it is <b>good enough to be useful</b>.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="../money/images/coin-stack-add.jpg" alt="coin stack add" height="140" width="106"></td>
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<td>
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<p>If we are charged by the volume of the box we send, we can take a few measurements and know how much to pay.</p>
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<p>It can also be useful when deciding which box to buy when we need to pack things.</p>
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<p class="center">So the model is useful!</p>
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</td>
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</tr>
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</tbody></table>
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<p> </p>
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<p>But maybe we need more accuracy, we might need to send hundreds of boxes every day, and the thickness of the cardboard matters. So let's see if we can <b>improve the model</b>:</p>
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<div class="example">
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<h3>The cardboard is "t" thick and we measure from outside the box. How much space is inside?</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/cardboard-box-b.svg" alt="Cardboard Box with thickness" height="130" width="110"></p>
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<p>The inside measurements need to be reduced by the thickness of each side:</p>
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<ul>
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<li>The inside length is <b>l-2t</b></li>
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<li>The inside width is <b>w-2t</b>,</li>
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<li>The inside height is <b>h-2t</b></li>
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</ul>
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<p>and now the formula is:</p>
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<p class="center"><span class="larger">Inside Volume = (l-2t) × (w-2t) × (h</span><span class="larger">-2t)</span></p>
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</div>
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<p>Now we have a <b>better</b> model. Still not perfect (did we consider wasted space because we could not pack things neatly, etc ...), but better.</p>
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<p class="center larger">So a model is not reality, but should be good enough to be useful.</p>
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<h2>Playing With The Model</h2>
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<p>Now we have a model, we can use it in different ways:</p>
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<div class="example">
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<h3>Example: Your company uses 200x300x400 mm size boxes, and the cardboard is 5mm thick.</h3>
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<h3>Someone suggests using 4mm cardboard ... how much better is that?</h3>
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<p>Let us compare the two volumes:</p>
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<ul>
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<li>Current Volume = (200-2×5) × (300-2×5) × (400-2×5) = 21,489,000 mm<sup>3</sup></li>
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<li>New Volume = (200-2×4) × (300-2×4) × (400-2×4) = 21,977,088 mm<sup>3</sup></li>
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</ul>
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<p>That is a change of:</p>
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<p class="center larger">(21,977,088-21,489,000)/21,489,000 ≈ 2% more volume</p>
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</div>
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<p>So the model is <b>useful</b>. It lets us know we will get 2% more space inside the box (for the same outside measurements).</p>
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<p>But there are still "real world" things to think about, such as "will it be strong enough?"</p>
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<h2>Thinking Clearly</h2>
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<p>To set up a mathematical model we also need to think clearly about the facts!</p>
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<div class="example">
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<h3>Example: on our street there are twice as many dogs as cats. How do we write this as an equation?</h3>
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<ul>
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<li>Let D = number of dogs</li>
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<li>Let C = number of cats</li>
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</ul>
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<p class="center">Now ... is that: <span class="large">2D = C</span></p>
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<p class="center">or should it be: <span class="large">D = 2C</span></p>
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<p>Think carefully now!</p>
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<p class="center">The correct answer is <span class="hide">D = 2C</span></p>
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<p> </p>
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<p>(<b>2D = C</b> is a common mistake, as the question is written "twice ... dogs ... cats")</p>
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</div>
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<p>Here is another one:</p>
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<div class="example">
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<h3>Example: You are the supervisor of 8-hour shift workers. They recently had their break times reduced by 10 minutes but total production did not improve.</h3>
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<p>At first glance there is nothing to model, because there was no change in production.</p>
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<p>But wait a minute ... they are working 10 minutes more, but producing the same amount, so <b>production per hour</b> must have dropped!</p>
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<p>Let us assume they used to work 7 hours (420 minutes):</p>
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<p class="center larger">Change in production per hour = 410/420 = <b>0.976...</b></p>
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<p class="center">Which is a <b>reduction of more than 2%</b></p>
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<p>But even worse: the first few hours of the shift are not affected by the shorter break time, so it could be a 4 or 5% reduction later in the shift.</p>
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<p>You could recommend:</p>
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<ul>
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<li>looking at production rates for every hour of the shift</li>
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<li>trying different break times to see how they affect production</li>
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</ul>
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</div>
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<p> </p>
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<h2>A Bigger Example: Most Economical Size</h2>
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<p>OK, let us have a go at building and using a mathematical model to solve a real world question.</p>
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<div class="center80">
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<p><b>Your company is going to make its own boxes!</b></p>
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<p>It has been decided the box should hold <b>0.02m<sup>3</sup></b> (0.02 cubic meters which is equal to 20 liters) of nuts and bolts.</p>
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<p>The box should have a square base, and double thickness top and bottom.</p>
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<p>Cardboard costs <b>$0.30</b> per square meter.</p>
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<p>It is up to you to decide the most economical size.</p>
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</div>
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<h3>Step One: Draw a sketch!</h3>
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<p>It helps to sketch out what we are trying to solve!</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/3d-square-box.svg" alt="3d Box" height="90" width="167"> </td>
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<td>The base is square, so we will just use "w" for both lengths</td>
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</tr>
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</tbody></table>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>The box has 4 sides, and double tops and bottoms.</p>
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<p>The box shape could be cut out like this (but is probably more complicated):</p>
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</td>
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<td><img src="images/3d-square-box-net.svg" alt="3d Box Net (top, sides, base)" height="177" width="301"></td>
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</tr>
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</tbody></table>
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<h3>Step Two: Make Formulas.</h3>
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<p>Ignoring thickness for this model:</p>
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<p class="center"><span class="larger">Volume = w × w × h = w<sup>2</sup>h</span></p>
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<p>And we are told that the volume should be 0.02m<sup>3</sup>:</p>
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<p class="center"><span class="larger">w<sup>2</sup>h = 0.02</span></p>
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<p>Areas:</p>
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<p class="center"><span class="larger">Area of the 4 Sides = 4 × w × h = 4wh</span></p>
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<p class="center"><span class="larger">Area of Double Tops and Bases = 4 × w × w = 4w<sup>2</sup></span></p>
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<p>Total cardboard needed:</p>
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<p class="center"><span class="larger">Area of Cardboard = </span><span class="larger">4wh + </span><span class="larger">4w<sup>2</sup></span></p>
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<h3>Step Three: Make a Single Formula For Cost</h3>
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<p>We want a single formula for cost:</p>
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<p class="center larger">Cost = $0.30 × Area of Cardboard</p>
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<p class="center"><span class="larger">= $0.30 × (4wh + </span><span class="larger">4w<sup>2</sup>)</span></p>
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<p>And that is the cost when we know width <b>and</b> height.</p>
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<p>That could be hard to work with ... a function with two variables.</p>
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<p>But we can make it simpler! Because width and height are already related by the volume:</p>
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<p class="center"><span class="larger">Volume = w<sup>2</sup>h = 0.02</span></p>
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<p class="center">... which can be rearranged to ...</p>
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<p class="center"><span class="larger">h = 0.02</span>/<span class="larger">w<sup>2</sup></span></p>
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<p class="center">... and that can be put into the cost formula ...</p>
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<p class="center"><span class="larger">Cost</span><span class="larger"> = $0.30 × (4w×<b>0.02/w<sup>2</sup></b> + </span><span class="larger">4w<sup>2</sup></span>)</p>
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<p>And now the cost is related directly to <b>width only</b>.</p>
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<p>With a little simplification we get:</p>
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<div class="center80">
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<p class="center"><span class="larger">Cost</span><span class="larger"> = $0.30 × (0.08/w+ </span><span class="larger">4w<sup>2</sup>)</span></p>
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</div>
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<h3>Step Four: Plot it and find minimum cost</h3>
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<p>What to plot? Well, the formula only makes sense for widths greater than zero, and I also found that for widths above 0.5 the cost just gets bigger and bigger.</p>
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<p>So here is a plot of that cost formula for <b>widths between 0.0 m and 0.55 m</b>:</p>
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<p class="center"><img src="images/model-box-cost-graph.gif" alt="Cost Graph" height="207" width="257"><br>
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<span class="larger">Plot of y= 0.3(0.08/x+4x<sup>2</sup>)<b><br>
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x</b> is width, and <b>y</b> is cost</span></p>
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<p>Just by eye, I see the cost reaches a minimum at about <b>(0.22, 0.17)</b>. In other words:</p>
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<ul>
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<li>when the width is about <b>0.22 m</b> (x-value),</li>
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<li>the minimum cost is about <b>$0.17</b> per box (y-value).</li>
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</ul>
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<p>In fact, looking at the graph, the width could be anywhere between 0.20 and 0.24 without affecting the minimum cost very much.</p>
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<h3>Step Five: Recommendations</h3>
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<p>Using this mathematical model you can now recommend:</p>
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<ul>
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<div class="bigul">
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<li>Width = 0.22 m</li>
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<li>Height = 0.02/w<sup>2</sup> = 0.02/0.22<sup>2</sup> = 0.413 m</li>
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<li>Cost = $0.30 × (0.08/w+ 4w<sup>2</sup>) = $0.30 × (0.08/0.22+ 4×0.22<sup>2</sup>) = $0.167</li>
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</div>
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</ul>
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<p>Or about 16.7 cents per box</p>
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<p>But any width between 0.20 m and 0.24 m is fine.</p>
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<p>You might also like to suggest improvements to this model:</p>
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<ul>
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<li>Include cost of glue/staples and assembly</li>
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<li>Include wastage when cutting box shape from cardboard.</li>
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<li>Is this box a good shape for packing, handling and storing?</li>
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<li>Any other ideas you may have!</li>
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</ul>
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<p> </p>
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<h2>Predicting the Future</h2>
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<p>Mathematical models can also be used to forecast future behavior.</p>
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<div class="example">
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<h3>Example: An ice cream company keeps track of how many ice creams get sold on different days.</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../combinatorics/images/ice-cream.jpg" alt="ice cream" height="133" width="100"></p>
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<p>By comparing this to the weather on each day they can make a mathematical model of <b>sales versus weather</b>.</p>
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<p>They can then predict future sales based on the weather forecast, and decide how many ice creams they need to make ... ahead of time!</p>
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</div>
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<h2>Computer Modeling</h2>
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<p>Mathematical models can get very complex, and so the mathematical rules are often written into computer programs, to make a computer model.</p>
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<p class="center larger">Have a play with a simple <a href="../geometry/ellipse-reflect-model.html">computer model of reflection inside an ellipse</a><br>
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or the <a href="../physics/pendulum.html">single pendulum</a> or <a href="../physics/double-pendulum.html">double pendulum animation</a>.</p>
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<p>More complex examples include:</p>
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<ul>
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<li>Weather prediction</li>
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<li>Economic Models (predicting interest rates, unemployment, etc)</li>
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<li>Public health vs infectious diseases</li>
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<li>Models of how large structures behave under stress (bridges, skyscrapers, etc)</li>
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