EXPLAIN your answers with DETAILS A homeowner has 80 feet of

EXPLAIN your answers with \"DETAILS\"

A homeowner has 80 feet of chain-link fencing to used to construct a dog pen adjacent to a house. (a) Express the area A(x) enclosed by the pen as a function of the width x. (b) Graph A(x) and determine the width x that will make the area maximum.

Solution

The chain link must cover the pen on 3 sides, thus length of these 3 sides is = x, x, 80-2x;
The area = length * breadth = x(80-2x) = 80x - 2x² ; A(x) = 80x - 2x² ; this graph will be a parabola passing through the origin;
For maximum area, we need to differentaite A(x) with respect to \'x\' and equate it to 0;
dA(x) / dx is 80-4x =0
Thus, 4x=80 or x=20;
x=20 ft will make the area maximum;

So for maximum area x=20 so width = 20 and length = 80-2*20 = 40 and maximum area is = 40*20 = 800 sq ft;