What is the maximum area for a rectangle with an perimeter of 120 meters?

Answer 1

Maximum area for a rectangle of a given perimeter is a square so side would be 30m and area 900 sq m

Math behind it:

We have two variables X and Y (width and height)

The perimeter of the rectangle is 120, therefore 2X + 2Y = 120

Now that we have an equation we can get rid of one of the variables like so:

2X + 2Y = 120 => 2Y = 120 - 2X => Y = 60 - X

We want to optimize the area (A), which is X*Y:

A = X * Y => Substitute the Y -> A = X * (60 - X) => A = 60X - X^2

To find the maximum area we'll have to find where the derivative A' zero.

A = 60X - X^2 => Find the derivative -> A' = 60 - 2X

A' = 0 => 60 - 2X = 0 => -2X = -60 => X = 30

Filling in the X to get the Y we get: Y = 60 - X => Y = 60 - 30 => Y = 30

Now that we have the X and the Y we can calculate the maximum area which is:

X * Y => 30 *30 => 900