Question

A farmer has 900 feet of fence to build a pen that is up against an existing wall, so fencing is only needed on three sides. What will be the dimensions of the pen that will maximize the area?

Answer 1

Assuming a rectangular fenced area, let the area measure $x$ feet by $y$ feet. Then the length of fencing used is such that

$2x + y = 900 \implies y = 900 - 2x$

where $y$ is the length of the side parallel to the existing wall.

The area enclosed by the fence is

$A(x,y) = xy \implies A(x) = x(900-2x) = 900x - 2x^2$

Find the critical points of $A(x)$.

$A'(x) = 900 - 4x = 0 \implies x = 225$

If two sides measure 225 feet, then the remaining side must measure

$y = 900 - 2\cdot225 = 450$

feet. So the pen must have dimensions 225-ft by 450-ft. We know this gives a maximum area, since by completing the square we find an upper bound for the enclosed area of

$900x - 2x^2 = -2 (x^2 - 450x) \\\\ ~~~~~~~~ = -2 (x^2 - 450x + 225^2 - 225^2) \\\\ ~~~~~~~~ = 2\cdot225^2 - 2 (x - 225)^2 \\\\ ~~~~~~~~ \le 2\cdot225^2 = 101,250$

square feet.