Answer:
A(max) = 1012,5 ft²
Dimensions:
x = 45 ft
y = 22,5 ft
Step-by-step explanation:
We have 90 ft of fence
Let call dimensions of rectangular garden x and y ( x will be the side running parallel to the wall) then
Area of rectangular garden is
A = x*y (1)
And perimeter of the rectangular area wich is P = 2*x * 2*y and as we will use fence in only one x side then
P = 90 = x + 2*y ⇒ y = ( 90 - x ) / 2
Then equation (1) becomes
A(x) = x* ( 90 - x ) / 2 ⇒ A(x) =( 90*x - x² ) / 2 ⇒ A(x) =45*x - x²/2
A(x) =45*x - x²/2
Taking derivatives on both sides of the equation
A´(x) = 45 - x
A´(x) = 0 ⇒ 45 - x = 0
x = 45 ft
And
y = ( 90 - x ) / 2 ⇒ y = ( 90 - 45 ) / 2
y = 22,5 ft
And th largest possible area is:
A(max) = x*y = 45*22,5
A(max) = 1012,5 ft²
