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: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*bTotal length of fence=L= 2*a+b thereforeL= 2*a+b → b= L- 2*aA= a*(L-2*a) = a*L - 2*a²thereforeA= 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=0thereforeL²- 8*A max=0 → A max = L²/ 8 = (200m)²/8= 5000 m²and since √(L²- 8*A)=0a=L/4 = 200m/4 = 50 mb= L- 2*a = 200m - 2* 50m = 100 m