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.
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 $
