Determine the dimensions of the rectangular that will enclose the largest area

Question

We need to enclose a rectangular field with a fence. We have 500 feet of fencing material; a building is on one side of the field and won’t need any fencing. Determine the dimensions of the field that will enclose the largest area?

Answer 1

maximize x*y

subject to 500 = x + 2*y

x=500-2*y

Area=A(y)=(500-2*y)*y=500y-2y²

Since the leading coefficient of the area function is negative and the function is quadratic, we know that the function has a unique maximum where the derivative of A(y)=0.

Compute the derivative of A(y), A'(y), and set it to 0.

A'(y)=500 - 4y

500 - 4y=0

500=4y

y=125

x=500 - 2*y

=500 - 2*125

=500 - 250

=250

So the dimensions of the field are x=250 and y=125.