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