Critical points and derivative

Question

A farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimension would require the least amount of fencing if no fencing is needed along the river? Show how you optimized by using critical points and the derivative.

Answer 1

180,000 square meters = A

A= LW = 180,000

Perimeter = L +2W

where W = distance to the river from the Longest side of the fence

L = the side of the fencing that's parallel to the river

L =2W

2W(W)= 180,000

W = 300 meters

L = 600 meters

300(600) = 180000

Minimize fencing by minimizing the Perimeter = L+2W

LW = 180,000

L = 180,000/W

Perimeter = P(W) = 180,000/W + 2W

take the derivative and set equal to zero

P'(W) = -180,000/W^2 +2 = 0

multiply by W^2

-180,000 +2W^2 = 0

W^2 = 180,000/2 = 90,000

W = sqr90000 = 300 meters wide

L = 180000/W = 180000/300 = 600 meters Long

L=600 m and W = 300 m minimizes the fencing to get 180,000 m^2 area