A farmer plans to enclose a rectangular pasture adjacent to a river (see figure). The pasture must contain 80,000 square meters in order to provide enough grass

Question

A farmer plans to enclose a rectangular pasture adjacent to a river (see figure). The pasture must contain 80,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing?

Answer 1

Let x = length of the side parallel to the river

      y = length of each of the other two sides

 

Then xy = 80,000, so y = 80,000/x

 

Minimize:  L = length of the fence

 

L = x + 2y

 

   = x + 160,000/x, where x > 0

 

L' = 1 - 160,000/x² = (x² - 160,000)/x²

 

L' = 0 when x = 400 or -400

 

Since x can't be negative, x must be 400.

 

When 0 < x < 400, L' < 0.  So, L is decreasing.

 

When x > 400, L' > 0.  So L is increasing.

 

Therefore, L has a relative and absolute minimum when x = 400 m

                                                                                     y = 80,000/x = 200 m

 

The fence is shortest if the side parallel to the river has length 400 m and the other 2 sides each have length 200 m.