Page 61 - Calc for the Clueless

Chapter 4
Word Problems Made Easy ... Well, Less Difficult

Word problems are not difficult because of the calculus. In fact, in more than 90% of word problems, the
calculus is very easy. The problem is that few precalculus courses properly prepare you (or prepare you at all)
for the setting up of the word problems. As you will see, the algebra is the difficult part of the problems.

The best we can do is to work out a number of word problems that are found in many calculus books. Many of
the techniques involved in these problems are applicable to other word problems.

max, min

The theory of these problems is simplicity itself: to find a maximum or a minimum, take the derivative, and set
it equal to 0.

In general, there is a picture to be drawn. Always draw the picture! Next we have to assign the variable or
variables in the problem. Hopefully, by doing enough good examples, you will see how this is done. (I will try
my best—you must try your best, which means don't panic.) Most of the problems will have two equations in
two unknowns. One of these is equal to a number. You will solve for one of the variables and substitute it in the
second equation. In the second equation you will take the derivative and set it equal to 0.

Let's start with an easy one.

Example 1—

A farmer wishes to make a small rectangular garden with one side against the barn. If he has 200 feet of fence,
find the garden of maximum area.

First we make what I call my crummy little picture. You only have to make the picture good enough so you can
understand what you drew.

Second, we assign variables. In this case, this task is easy. The two equations involve the area and the
perimeter. A = xy. p = 2x + y. The trick, if you could call it a trick, is that the barn is one side of the rectangle.
The perimeter involves counting y once.

The area is x times y, which is 5000 square feet. Sometimes the problem only asks for the dimensions, in which
case 50 feet and 100 feet are the answers.

To see whether this is truly a max, we find A"(50). Since A" = -4, x = 50 is a maximum. We will check only a
few times. Sometimes it is too messy. Usually it just takes more time than I feel like taking.

Example 2—