Amanda's calculus course is progressing. I am progressing, too--through Martin Gardner's update of Silvanus Thompson's *Calculus Made Easy*, and it's making me nostalgic. I remember how, after first learning to differentiate functions, we had our first word problems, and how cool I thought it was to be able to solve, so easily, very practical problems that would be impossible were it not for the calculus. For example, I can still recall one of the problems from the second exam in Calculus I:

You want to build a storage shed that is to have a square base, a flat roof, and a volume of 1800 cubic feet. Find the dimensions of the least expensive shed that can be built if material for the floor costs $1.20 per square foot, for the sides $3.00 per square foot, and for the roof $2.00 per square foot.

So there is a cost function, which we want to minimize. The cost is the sum of certain areas multiplied by the cost of the material for that area, or

C = 1.2x

^{2}+ (3.00)(4)xh + 2.00x^{2}

The two variables are a potential road block but, as the shed is to have a volume of 1800 cubic feet, the height (h) can be expressed in terms of the dimension of the square base (x), or 1800/x^{2}. Now, combining like terms and substituting, we have

C = 3.2x

^{2}+ (12x)(1800/x^{2}) = 3.2x^{2}+ 21,600/x

This expression will have a minimum value when its derivative--here comes the calculus--is equal to zero. So, differentiating, we get

dC/dx = 6.4x - 21,600/x

^{2},

which is equal to zero when

6.4x = 21,600/x

^{2}.

Doing the algebra, we get

6.4x

^{3}= 21,600: x^{3}= 21,600/6.4: x = 3375^{1/3}= 15.

If the square base is 15 feet on a side, and the volume 1800 cubic feet, then the height has to be 1800/225, or 8. The least expensive shed that can be built is 15 feet wide by 15 feet deep by 8 feet high.

Now calculus, like, say, "Shakespeare," has an undeserved mystical aura among subjects in the school curriculum. We just got the right answer to what might have seemed to be quite a difficult problem. All that was needed, besides a bit of algebra, was the power rule for derivatives--more like a straightforward technique than anything needing deep, mystic knowledge.

If you are as big a dweeb as I you might enjoy this and this, among others.

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