find the dimensions of the maximum rectangular area that can be made with 900m of fence.

Question

the city park department is planning an enclosed play area in a new park. One side of the area will be against and existing building, with no fence needed there. find the dimensions of the maximum rectangular area that can be made with 900m of fence. 

Answer 1

Let x = length of each of the two sides perpendicular to the wall

Let y = length of the side parallel to the wall

 

Then 2x + y = 900 (that is how much fence we have); so y = 900 - 2x

 

The area of the rectangle is: A = xy = x (900 - 2x) = 900x - 2x²

 

Since this is a parabola opening downward we could locate the vertex to find the x value that generates a maximum area. Or we could find A' and set equal to zero, solving for x to find the value that generates the max.

 

A' = 900 - 4x; when A' = 0, x = 225. When x = 225 y = 450. So the rectangle that generates a max aree is 225 x 450.

Answer 2

The rectangle with the maximum area for a given perimeter is the square.

The park should be 225m x 225m.