You need to enclose a 250 square foot section with a fence that costs $1.50 per yard. what dimensions would you use to minimize the cost.

Width in yards = w
Lenght in yards =
l

The total area is 250 square
feet.

Fence costs $1.50 per yard or $0.50 per
foot.

Total area = w*l = 250 square feet.
(1)

Perimeter = 2w + 2l =
2(w+l)

Cost = Perimeter in feet times cost per
foot.
C = 0.50(2(w + l) = 1(w+l) = w+l (2)

From
(1) l = 250/w and we substitute into (2) to get

C = w +
250/w   We want to minimize this, so take the
derivative.

C' = 1 - 250/w^2   C will be at a minimum or
maximum when C' = 0.

1 - 250/w^2 = 0     Solve for
w.

1 = 250/w^2

w^2 =
250

w = 50 feet  (we reject the negative square root
because width is always > 0.

if w = 50 then l = 50 
since w*l = 250.   Cost = (50+50) = $100