Isaac wants to enclose his backyard with a fence that is bounded on one side by the back of the house. if he has 164 feet of fencing material find the maximum area that can be enclosed by the fence.

Answer 1

let the side parallel to the house be y ft
let each of the other two sides by x ft
so
2x + y = 164
y = 164 - 2x

area = xy
= x(164 - 2x
= -2x^2 + 164x

This is a downwards opening parabola, so we need its vertex.
Either
1. complete the square
2. use Calculus
3. use " the x of the vertex is -b/(2a) for any y = ax^2+bx+c

Iwill use the third method:
x of the vertex = -164/-4 = 41
then y = 164-82 = 82

the maximus area = xy = (41)(82) = 3362 square ft
when the field is 41 by 82 ft