Question

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.

Answer 1

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