John has 300 feet of lumber to frame a rectangular patio the

John has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a rectangle is length times width). What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation.

Solution

Perimeter =2l+2w=300 ft.
Dividing by 2 you get:
l+w=150
Area=(length)(width)
Let length be \"l\"
Then width = \"150-l\"
Area=(l)(150-l)
Area=150l-l^2
This is a quadratic equation where a=-1 and b=150.
The maximum area is where l=-b/2a=-150/(-2)=75
So the maximum area is as follows:
Area=lw=(75)(150-75)=75(75)=5625 sq. ft.