You have 9090 ft of fence to make a rectangular vegetable garden alongside the wall of your house. The wall of the house bounds one side of the vegetable garden. What is the largest possible area of the vegetable garden?

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