High School

A total of 2000 square feet is to be enclosed in two pens, as illustrated. The outside walls are to be constructed of brick, and the inner dividing wall is to be constructed of chain link. The brick wall costs $10 per linear foot, and the chain link costs $5 per linear foot.

Find the dimensions [tex]x[/tex] and [tex]y[/tex] that minimize the cost of construction.

Answer :

The dimensions x and y that minimize the cost of construction is 22.3 ft x 89.9 ft

How to find the Dimensions that minimize the cost of construction

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 $10 per linear foot and the chain link costs $5 per linear foot. Thus;

C(x) = 10x + 5y

From eq(1),y = 2000/x

Thus;

C(x) = 20x + 5(2000/x)

C(x) = 20x + 10000/x

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

Thus;

C'(x) = 20 - 10000/x²

Equating to zero;

20 - 10000/x² = 0

20 = 10000/x²

20x² = 10000

Divide both sides by 20;

x² = 8000/20

x² = 500

x = √500

x = 22.3 ft

Putting 22.3 for x in eq 1,we have;

20y = 2000

y = 2000/22.3

y = 89.9 ft

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