Question

A fence is to be built to enclose a rectangular area of 320 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 12 dollars per foot. Find the length L and width W (with W≤L) of the enclosure that is most economical to construct.

Answer 1

You neglect to say whether the $15/ft section must be built along the width or the length

Chooing W (since it is smaller than L), we have

(1) LW = 320

(2) 5(L + L) + 5W + 15W = C (cost)

or

(3) C = 10L + 20W

Using (1), L = 320/W, subing into (3):

C = 3200/W + 20W


To minimize C, dC/dW = 0 = -3200/(W^2) + 20

Solving for W:

W = sqrt(3200/20) = 12.649ft.

From which we get L = 25.298ft.