A fence must be built to enclose a rectangular area of 20,000 square feet. fencing material cost $2.50 per foot for the two sides facing north and south and $3.50 per foot for the other two sides. find the cost of the least expensive fence.

The area is:

A = x * y = 20000

The cost function is:

C = 2.50 (2x) + 3.50 (2y)

Rewriting we have:

C = 5x + 7y

Writing as a function of x we have:

C = 5x + 7 (20000 / x)

Rewriting:

C (x) = 5x + 140000 / x

We derive:

C ' (x) = 5-140000 / x ^ 2

We equal zero and clear x:

0 = 5-140000 / x ^ 2

140000 / x ^ 2 = 5

x ^ 2 = 140000/5

x = root (140000/5)

x = 167.33 feet

Therefore the cost is:

C (167.33) = 5 * (167.33) + 7 * (20000 / 167.33)

C (167.33) = 1673.32 $

Answer:

The cost of the least expensive fence is:

$ 1673.32