For tax reasons, I need to create a rectangular vegetable patch with an area of exactly 32 sq. ft. The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. What are the dimensions of the vegetable patch with the least expensive fence?

north and south sides

ft

east and west sides

ft

Question

Ansley Woodward

Mathematics

5 November, 11:27

For tax reasons, I need to create a rectangular vegetable patch with an area of exactly 32 sq. ft. The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. What are the dimensions of the vegetable patch with the least expensive fence?

north and south sides

ft

east and west sides

ft

+5

Answers (1)

Know the Answer?

Briana Huerta · Answer 1

Let the length of the north and south sides be x, then the length of the east and west side is 32/x.

The total fencing needed is the perimeter of the rectangle = 2x + 2 (32/x) = 2x + 64/x

Cost of fencing the north and south sides is 2 (2x) = 4x

Cost of fencing the east and west sides is 4 (64/x) = 256/x

Total cost (C (x)) of fencing the vegetable patch is 4x + 256/x

For minimum cost dC/dx = 0

4 - 256/x^2 = 0

4x^2 - 256 = 0

x^2 = 64

x = 8

Therefore, the dimensions of the vegetable patch with the least expensive fence is

north and south sides = 8 ft

east and west sides = 32/8 = 4 ft