Question

A farmer needs to fence in a rectangular plot of land, and he has 200 meters of fence to work with. He is going to construct the plot next to a river, so he will only have to use fence for three sides of the plot. Find the dimensions that will allow the farmer to maximize the area of the plot.

Answer 1

the farmer will need to have 2 pieces of 50 m and one of 100m to maximise the area ( Maximum area=5000 m²)

Explanation:

if we assume that the southern fence is not required and if we denote a= length of each lateral side and b= length of the front side , we will have:

Area=A= a*b

Total length of fence=L= 2*a+b

therefore

L= 2*a+b → b= L- 2*a

A= a*(L-2*a) = a*L - 2*a²

therefore

A= a*L - 2*a² → 2*a² - a*L + A = 0

a= [L ± √((-L)²- 4*2*A)]/(2*2) = L/4 ± √(L²- 8*A) /4

a= L/4 ± √(L²- 8*A) /4

when A goes bigger √(L²- 8*A) diminishes, but since the minimum possible value √(L²- 8*A) is 0 , then A can not go higher than L²- 8*A=0

therefore

L²- 8*A max=0 → A max = L²/ 8 = (200m)²/8= 5000 m²

and since √(L²- 8*A)=0

a=L/4 = 200m/4 = 50 m

b= L- 2*a = 200m - 2* 50m = 100 m