A fence must be built to enclose a rectangular area of 45 comma 000 ftsquared. fencing material costs $ 1 per foot for the two sides facing north and south and $2 per foot for the other two sides. find the cost of the least expensive fence.

The area is:

A = x * y = 45000 feet ^ 2

The cost function is given by:

C = 1 * (2x) + 2 * (2y)

We write the function in terms of x:

C (x) = 1 * (2x) + 2 * (2 (45000 / x))

Rewriting we have:

C (x) = 2x + 180000 / x

We derive the expression:

C ' (x) = 2 - 180000 / x ^ 2

We match zero:

2 - 180000 / x ^ 2 = 0

We clear x:

2 = 180000 / x ^ 2

x ^ 2 = 180000/2

x = root (90000)

x = 300 feet

Therefore the total cost will be:

C (300) = 2 * (300) + 180000/300

C (300) = 1200 $

Answer:

The cost of the least expensive fence is:

C (300) = 1200 $