Question in the description

Question

A fence is to be built to enclose a rectangular area of 250 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 14 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Answer 1

Let x = length of a side costing $3 per foot and let y = length of the side costing $14 per foot.

xy = 250, so y = 250/x.

Total cost = C = 3(2x + y) + 14y = 6x + 17y = 6x + 4250/x

dC/dx = 6 - 4250 / x² = (6x² - 4250) / x². The cost will be minimized when dC/dx = 0.

6x² = 4250

x² = 708.33333

x = 26.61 ft and y = 9.39 ft