Question

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.

Answer 1

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 $