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A rectangular lettuce patch, 560 square feet in area, is to be fenced off against rabbits. Find the least amount of fencing required if one side of the land is already protected by a barn. What are the dimensions?

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raphealnwobi

The dimensions for the least amount of fencing is 16.7 ft by 33.5 ft

Let x represent the length of the fence and y represent the width of the fence.

Since the area is 560 square feet, hence:

Area = length * width

560 = xy

y = 560/x

One side of the land is already protected by a barn, hence:

Perimeter (P) = x + x + y = 2x + y

P = 2x + y

P = 2x + 560/x

For least amount of fencing, dP/dx = 0, hence:

dP/dx = 2 - 560/x²

0 = 2 - 560/x²

560/x² = 2

2x² = 560

x = 16.7

y = 560/16.7 = 33.5 ft

The dimensions for the least amount of fencing is 16.7 ft by 33.5 ft

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