Finding a length

Question

A farmer wants to fence a rectangular area of 50 sq ft next to a river. Find the length and width of the rectangular which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to rhe river.

Answer 1

5 feet wide 10 feet long

for a total of 20 feet total fencing

any other dimensions require more fencing for an area of 50 feet

area=LW = L(50/L)=50

Perimeter = P = 2(50/L) + L = 100/L+L

take the derivative and set =0

P' = -100/L^2 + 1 = 0

multiply by L^2

L^2 -100 =0

L=10 feet long

W =50/10 = 5 feet wide