Fence Dimensions

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 5 dollars per foot, and the material for the fourth side costs 15 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Answer 1

This is an optimization question that we can solve as follows:

Let x: width of enclosure (both sections of width cost $5/ft)

Let y: length of enclosure (one section of length costs $15/ft)

Cost = 5·2x + 5y + 15y = 10x + 20y

Constraint: xy = 320 ---> y = (320/x)

Substitute this expression in for x in the cost function to get cost as a function of x only:

C(x) = 10x + 20(320/x) = 10x + 6400/x = 10x + 6400x^-1

C^'(x) = 10 - 6400x^-2 = 0

10x² - 6400 = 0

x = 8√10 , y = 4√10