Question

A rectangular area adjacent to a river is to be enclosed by a fence on the other three sides. If the area to be enclosed is 200 m2 and if no fencing is needed along the river, what dimensions require the least amount of fencing

Answer 1

Dimension is 20m x 10m

Explanation:

The area of the rectangular is 200 ft².

Let;

x = length of the side parallel to the river

y = length of each of the other two sides

Then xy = 200, so y = 200/x

Minimize: L = length of the fence

L = x + 2y

Thus,

L = x + 400/x, where x > 0

dL/dx = 1 - 400/x² = (x² - 400)/x²

Let dL/dx =0

Thus, x² - 400 = 0

x = ± 20

Since x can't be negative, x must be 20.

When 0 < x < 20, dL/dx < 0. So, L is decreasing.

When x > 20, dL/dx > 0. So L is increasing.

Therefore, L has a relative and absolute minimum when x = 20 m

y = 200/x = 200/20 = 10 m

The fence is shortest if the side parallel to the river has length 20 m and the other 2 sides each have length 10 m.