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.

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

Step-by-step 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