What is the total area?

Question

a person plans to use 60 feet of fencing to enclose a rectangular patio. What dimensions of the rectangle would give the maximum area? what is the area?

Answer 1

So what do we know. Let L be length and W be width.

Perimeter of a rectangle 

P = 2L+2W = 60

Thus L = 30-W

 

The area of a rectangle is

A = LW

 

Plug in for L

A = (30-W)W = 30W-W²

 

Take derivative wrt W and set to 0. 

dA/dW = 30-2W = 0

Thus W = 15

Take second derivative to see if max or min

d²A/dW²= -2 

So our critical point is max sisince second derivative is negative.

 

So we know L = 30-W=15.

So our dimensions are 15x15 with max area being 225. In fact a square

Graph A = 30W-W² and you should see this.

 

Hope helpful. Best - David

Answer 2

For a given perimeter the rectangle with the greatest area is the square.

The dimensions are 15 x 15, and the maximum area is 225.