A fence is to be built to enclose a rectangular area of 1250 square feet. the fence along three sides is to be made of material that costs $3 per foot. the material for the fourth side costs $9 per foot. find the dimensions of the rectangle that will allow for the most economical fence to be built.

Let x and y be the length and width of the rectangle.

Because the area is 1250 ft², therefore

xy = 1250, or

y = 1250/x

We know that three sides cost $3 per foot, and the fourth side costs $9 per foot.

Case 1: Three sides are (x, y, x) and the 4-th side is y.

The cost is

C = 3 (2x + y) + 9y

= 6x + 12y

= 6x + 12 (1250/x)

= 6x + 15000/x

For C to be minimum, C' = 0. That is

6 - 15000/x² = 0

x² = 15000/6

x = 50 ft, y = 1250/50 = 25 ft

C = 6 (50) + 15000 / (50²) = $600

Case 2: Three sides are (y, x, y) and the 4-th side is x.

The cost is

C = 3 (x + 2y) + 9x

= 12x + 6y

= 12x + 6 (1250/x)

= 12x + 7500/x

For C to be minimum, C' = 0.

12 - 7500/x² = 0

x² = 7500/12 = 625

x = 25 ft, y = 50ft

C = 12 (25) + 7500/25 = $600

Answer: 25 ft by 50 ft