Find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 264 meters

Question

Find the length and width of a rectangle that has the given perimeter and a maximum area.
Perimeter: 264 meters

Answer 1

Let width = w

Let length = x

 

You will need to use these equations in the optimization:

 

2(x + w) = 264

 

x + w = 132                    eq1

 

Area = xw                       eq2

 

You want to find the maximum area, so substitute the perimeter equation (eq1) into the area equation (eq2).  This will get the area equation in terms of only x.

 

Area = x(132 - x)

 

Area = 132x - x²

 

 

Then, take the derivative of Area and set it equal to zero.

 

d/dx (Area) = 0

 

132 - 2x = 0

 

Solve for x to get the critical value.  This critical value is the length that will give the maximum area.  Then once you have the length, plug it into eq1 to get the width.