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Applications of Differential Equations

# Applications of Differential Equations - CHAPTER 3 BASIC...

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CHAPTER 3. BASIC MATHEMATICAL MODELLING 3.1. What is Mathematical Modelling? Mathematical Modelling is the art of using mathematics to analyse SIMPLE situa- tions which are supposed to approximate VERY COMPLICATED realistic situations. Some students find it a bit hard to understand the idea of modelling, so let’s begin with some examples where the maths is trivial! Then we can focus on the modelling bit. Eventually we want to use our knowledge of differential equa- tions to set up models, but there won’t be any 1

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differential equations in the first few examples. How should CANS be made? Suppose you are an engineer helping to turn a factory making “tin” cans. Your objective is to MINIMIZE the cost of this process. Now IN REALITY the function C (), which gives you the cost of making one can, depends on a large number of things: the price of “tin”, the shape of the can, cost of sticking its parts together, etc... Very complicated. So let’s start with the SIMPLEST POSSIBLE MODEL: Model 1 In THIS [very simple] model, we only care about the amount of “tin” actually in the CAN. 2
h r If the height of the can is h and its radius is r , you can easily see that the area is: A = 2 πr 2 + 2 πrh This is minimized by setting r = 0... the cheap- est way to make cans IS NOT TO MAKE THEM AT ALL! TRUE! oh, OK, you want to put something inside those cans! 3

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V = πr 2 h = CONSTANT h = V πr 2 A ( r ) = 2 πr 2 + 2 V r A 0 = 0 4 πr - 2 V r 2 = 0 πr 2 h = V = 2 πr 3 h = 2 r We see that, when we impose the condition that the volume should be constant, the area becomes a function of r, as shown in the graph. In the graph, we chose h = 1, and you can see that the minimum is indeed, as calculus shows, at r = 1/2: the radius should be half the height, 4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 20 40 60 80 100 120 r A that is, the diameter should EQUAL the height. SO CANS SHOULD EITHER NOT BE MADE OR THEY SHOULD ALWAYS BE EXACTLY AS HIGH AS THEY ARE WIDE But can-manufacturers don’t usually do this, except for LARGE cans, like cans of paint! So our model is predicting something wrong 5

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WE NEED A MORE COMPLEX MODEL Model 2 As with model 1, BUT also we care about WASTAGE. The top and bottom of the can are punched out of flat metal, perhaps in the way shown. YOU HAVE TO PAY for the whole sheet! Well actually you should not punch the holes in this way, but rather in this way instead: But still you have to pay for a whole HEXAGON, as shown: An exercise in trigonometry should convince 6
r you that the area of the hexagon shown is 2 3 r 2 . So the amount of “tin”we have to pay for is not 2 πr 2 but rather 2 × 2 3 r 2 (multiply by 2 be- cause we need to make the top and bottom). A = 4 3 r 2 + 2 V r 7

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and now πr 2 h = V = 4 3 r 3 h = 4 3 π r instead of 2 r BUT 4 3 π 2 . 21 > 2. So our NEW MODEL SAYS THAT CANS SHOULD BE ABOUT 10 % HIGHER THAN WIDE... BETTER!
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Applications of Differential Equations - CHAPTER 3 BASIC...

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