Question

How would I solve this problem?

Answer 1

The area of the garden enclosed by the fencing is

A(x, y) = xy

and is constrained by its perimeter,

P = x + 2y = 200

Solve for x in the constraint equation:

x = 200 - 2y

Substitute this into the area function to get a function of one variable:

A(200 - 2y, y) = A(y) = 200y - 2y²

Differentiate A with respect to y :

dA/dy = 200 - 4y

Find the critical points of A :

200 - 4y = 0   ⇒   4y = 200   ⇒   y = 50

Compute the second derivative of A:

d²A/dy² = -4 < 0

Since the second derivative is always negative, the critical point is a local maximum.

If y = 50, then x = 200 - 2•50 = 100. So the farmer can maximize the garden area by building a (100 ft) × (50 ft) fence.