A rectangular lot adjacent to a highway is to be enclosed by a fence. If the fencing costs $2.50 per foot along the highway and $1.50 per foot on the other sides, find the dimensions of the largest lot that can be fenced off for $720.

Question

A rectangular lot adjacent to a highway is to be enclosed by a fence. If the fencing costs $2.50 per foot along the highway and $1.50 per foot on the other sides,  find the dimensions of the largest lot that can be fenced off for $720.

oobleck · Accepted Answer

Let the highway side have length x, and the rest will have total length x+2y.
We know that the cost is such that
2.50x + 1.50(x+2y) = 720
4x + 3y = 720
so, y = 240 - 4/3 x
The area a = xy = x(240 - 4/3 x) = 240x - 4/3 x^2
da/dx = 240 - 8/3 x
da/dx=0 when x = 90
So, the maximum area is 90 * 120 = 10800 ft^2