Page 65 - Calc for the Clueless

This problem is very similar to the minimizing of the cost of the fencing in Example 2, part B (the
multiplication part). To become good at these problems, you must notice similarities in problems and use
similar techniques in the problems. Then and only then will you become dynamite in these problems. Also note
that no matter how good you get, there will always be some problems that will give you trouble.

Example 6—

A printer is to use a page of 108 square inches with 1-inch margins at the sides and bottom and a ½-inch margin
at the top. What dimensions should the page be so that the area of the printed matter will be a maximum?

The area of the whole page is xy = 108. So y = 108/x. The area of the printed matter A = (x - 2)(y - 1.5) = xy -
1.5x - 2y + 3. Substituting y = 108/x, we get

The page should be 12 by 9 to have the largest amount of print.

Example 7—

The strength of a rectangular beam varies jointly as the width and the square of its depth. Which rectangular
beam that can be cut from a circular log of radius 10 inches will have maximum strength?

2
If we let x be the width and y be the depth, we can write the equation without a picture. The strength S = kxy ; k
is an unknown constant. To find a relationship between x and y, we need a picture of the log.

2
One of the things we always look for is the Pythagorean relationship. In this case x + y = 400 (the square of
2
2
the diameter). In the original equation, it is easier to solve for y , because if we solved for x we would have a
square root, which would make the derivative much more difficult and sometimes impossible to finish.
Therefore