Question

A total of 2,000 square feet is to be enclosed in two pens, separated by a chain link fence. The outside walls are to be constructed of brick. The brick wall costs $20 per linear foot and the chain link costs $4 per linear foot. Find the dimensions that minimize the construction cost

Answer 1

Dimensions that minimize is; 20 ft x 100 ft

Explanation:

Let the width and length be x and y respectively.

We are given area as 2000 Sq.ft.

Thus;

xy = 2000 - - - (eq 1)

We are told that the brick wall costs $20 per linear foot and the chain link costs $4 per linear foot. Thus;

C(x) = 20x + 4y

From eq(1),y = 2000/x

Thus;

C(x) = 20x + 4(2000/x)

C(x) = 20x + 8000/x

To minimize this, we will differentiate and equate to 0.

Thus;

C'(x) = 20 - 8000/x²

Equating to zeeo;

20 - 8000/x² = 0

20 = 8000/x²

20x² = 8000

Divide both sides by 20;

x² = 8000/20

x² = 400

x = √400

x = 20 ft

Putting 20 for x in eq 1,we have;

20y = 2000

y = 2000/20

y = 100 ft