# An arched window formed from the union of a semicircle and a rectangle has total perimeter #540"cm"#. What is the maximum possible area of the window?

##### 1 Answer

Maximum area is achieved when the bounding rectangle is a square of side

#### Explanation:

Let the length of the base of the rectangle be

Then the length of the perimeter of the semicircle is

The total perimeter of the window is

So

So

The area of the rectangle is

The area of the semicircle is

So the total area of the window is

This has maximum value when

The total height of the window is

Basically the maximum area is achieved when the bounding rectangle is a square of side