Rectangles are inscribed inside a semicircle of radius r. Find the rectangle with maximum area.

Explanation

To find the rectangle with the maximum area inscribed in a semicircle of radius \( r \), we can follow these steps:

### Step 1: Define the variables
Let the width of the rectangle be \( 2x \) (since it is symmetrical about the y-axis) and the height be \( y \).

### Step 2: Use the Pythagorean theorem
Since the rectangle is inscribed in a semicircle, we can use the Pythagorean theorem. The relationship between \( x \), \( y \), and the radius \( r \) of the semicircle is given by:
\[
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