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*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
