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
