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# 13.2 Optimization Problems.

13.8   Example. A stick of length is to be broken into four pieces of length and and the pieces are to be assembled to make a rectangle. How should and be chosen if the area of the rectangle is to be as large as possible? What is the area of this largest rectangle? Before doing the problem you should guess the answer. Your guess will probably be correct.

Let be the length of one side of the rectangle. Then so ; i.e., is a function of . Let be the area of a rectangle with side . Then

I include the endpoints for convenience; i.e., I consider rectangles with zero area to be admissible candidates for my answer. These clearly correspond to minimum area. Now

so has only one critical point, namely , and

Since is continuous on we know that has a maximum and a minimum, and since is differentiable on the extreme points are a subset of . Since the maximal area is ; i.e., the maximal rectangle is a square. (As you probably guessed.)

This problem is solved by Euclid in completely geometrical terms [17, vol 1 page 382].

Euclid's proof when transformed from geometry to algebra becomes the following. Suppose in our problem , say . Since , it follows that (if and were both less than , we'd get a contradiction, and if they were both greater than , we'd get a contradiction). Let be defined by

Then so

Hence, if , and to get a maximum we must have and . This proof requires knowing the answer ahead of time (but you probably were able to guess it). In any case, Euclid's argument is special, whereas our calculus proof applies in many situations.

Quadratic polynomials can be minimized (or maximized) without calculus by completing the square. For example, we have

From this we can easily see that for all and equality holds only if . This technique applies only to quadratic polynomials.

13.9   Example. Suppose I have 100 ft. of fence, and I want to fence off 3 sides of a rectangular garden, the fourth side of which lies against a wall and requires no fence (see the figure). What should the sides of the garden be if the area is to be as large as possible?

This is a straightforward problem, and in the next exercise you will do it by using calculus. Here I want to indicate how to do the problem without calculation. Imagine that the wall is a mirror, and that my fence is reflected in the wall.

When I maximize the area of a garden with a rectangle of sides and , then I have maximized the area of a rectangle bounded by 200 feet of fence (on four sides) with sides and . From the previous problem the answer to this problem is a square with . Hence, the answer to my original question is , . Often optimization problems have solutions that can be guessed on the basis of symmetry. You should try to guess answers to these problems before doing the calculations.

13.10   Exercise. Verify my solution in the previous example by using calculus and by completing the square.

13.11   Example. I want to design a cylindrical can of radius and height with a volume of cubic feet ( is a constant). How should I choose and if the amount of tin in the can is to be minimum?

Here I don't see any obvious guess to make for the answer.

I have

so . Let be the surface area of the can of radius . Then

The domain of is . It is clear that , and . Now . The only critical point for is (call this number ). Then . We see that for and for so is decreasing on and is increasing on and thus has a minimum at . The value of corresponding to is

Thus the height of my can is equal to its diameter; i.e., the can will exactly fit into a cubical box.

In the following four exercises see if you can make a reasonable guess to the solutions before you use calculus to find them.

13.12   Exercise. A box (without a lid) is to be made by cutting 4 squares of side from the corners of a square, and folding up the corners as indicated in the figure.
How should be chosen to make the volume of the box as large as possible?

13.13   Exercise. A rectangular box with a square bottom and no lid is to be built having a volume of 256 cubic inches. What should the dimensions be, if the total surface area of the box is to be as small as possible?

13.14   Exercise. A Find the point(s) on the parabola whose equation is that are nearest to the point .

13.15   Exercise. A Let and let . Find the point(s) r on the -axis so that path from to r to q is as short as possible; i.e., such that length([p r]) length([r q]) is as short as possible.
You don't need to prove that the critical point(s) you find are actually minimum points.

Next: 13.3 Rates of Change Up: 13. Applications Previous: 13.1 Curve Sketching   Index
Ray Mayer 2007-09-07