Area of a fenced garden with one side against the house so 3 sided

Question

A homeowner has 64 feet of fence to enclose his garden. Only 3 sides of the garden need to be fenced because one side of garden is against the house. What dimensions will maximum the area?

Answer 1

Let y = length of side parallel to the side with no fence

     x = length of each of the other two sides

 

Then, y+2x = 64.  So, y = 64-2x

 

Area = A = xy = x(64-2x)

 

The graph of the area function is a parabola opening downward with a highest point and with x-intercepts (0,0) and (32,0).

 

By the symmetry of the parabola, the x-coordinate of the maximum point lies halfway between the x-intercepts.  

 

So, the area is maximized when x = 16 ft

                                             y = 64 - 2x = 32 ft

 

Maximum area = xy = (16)(32) = 512 ft²

 

To maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.   

 

 

Answer 2

For the fixed given perimeter, the maximum area which can be obtained is a square.

So for 3 sides= 64 feet

one side = 64/3

The maximum area is 64/3*64/3 square feet