Page 103 - Calculus Demystified

CHAPTER 3 Applications of the Derivative
90
EXAMPLE 3.9
A rectangular garden is to be constructed against the side of a garage.The
gardener has 100 feet of fencing, and will construct a three-sided fence;
the side of the garage will form the fourth side. What dimensions will give
the garden of greatest area?
SOLUTION
Look at Fig. 3.14. Let x denote the side of the garden that is perpendicular
to the side of the garage. Then the resulting garden has width x feet and length
100 − 2x feet. The area of the garden is
2
A(x) = x · (100 − 2x) = 100x − 2x .

x
garage

_
100 2x
x
Fig. 3.14

We calculate A (x) = 100 − 4x and find that the only critical point for the
problem is x = 25. Since A (x) =−4 for all x, we determine that x = 25 is

a local maximum. By inspection, we see that the graph of A is a downward-
opening parabola. So x = 25 must also be the global maximum that we seek.
The optimal dimensions for the garden are
width = 25 ft. length = 50 ft.

You Try It: Find the right circular cylinder of greatest volume that can be
contained in a sphere of radius 1.
EXAMPLE 3.10
The sum of two positive numbers is 60. How can we choose them so as to
maximize their product?
SOLUTION
Let x be one of the two numbers. Then the other is 60 − x. Their product is
2
P(x) = x · (60 − x) = 60x − x .