Afence must be built to enclose a rectangular area of 20 comma 000 ftsquared. fencing material costs $ 2 per foot for the two sides facing north and south and $4 per foot for the other two sides. find the cost of the least expensive fence.

The first thing we should do is write the functions.
 For the area we have:
 A = w * h = 20000 feet ^ 2
 The price is:
 Cost = (4 * 2 * h + 2 * 2 * w)
 Substitute for h ::
 C (w) = (4 * 2 * (20,000 / w) + 2 * 2w)
 Rewrite we have:
 C (w) = 160,000 / w + 4w
 Take the derivative:
 C '(w) = 160000 (-1 / w ^ 2) + 4
 We equal zero and clear w:
 0 = 160000 (-1 / w ^ 2) + 4
 160000 / w ^ 2 = 4
 w ^ 2 = 40000
 w = 200 (approximately)
 width = 200 ft; so cost is 2 * ($ 4) 200 = $ 1600
 height = 20000/200 = 100 ft; so cost is 2 ($ 2) 100 = $ 400
 Total cost = $ 2000
 
 The cost of the least expensive fence is:
 Total cost = $ 2000