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

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