A homeowner wants to build a fence to enclose a 80 square yard rectangular area in his backyard. Along one side the fence is to be made of heavy-duty material costing $9 per yard, while the material along the remaining three sides costs $1 per yard. Determine the least cost to the homeowner.

L = 4 yard, W = 20 yard

Step-by-step explanation:

Area of backyard, A = 80 sq yard

Let the length of the fence is L and the width is W.

The cost of one side is $ 9 per yard and the cost of three sides is $ 1 per yard.

A = L x W = 80

W = 80 / L     .... (1)

Total cost, C = L x 9 + L x 1 + 2 W x 1

C = 10 L + 2 W

C = 10 L + 2 (80 / L)    from equation (1)

For maxima and minima differentiate C with respect to L and put it equal to zero.

dC/dL = 0

L² = 16

L = 4 yard

from equation (1)

W = 80 / 4 = 20 yard

SO, the cost is minimum is the length is 4 yard and the width is 20 yard.

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