A rectangle is inscribed in a semi-circle of radius `r` with one of its sides on diameter of semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find also the area.

Explanation

To solve the problem of finding the dimensions of a rectangle inscribed in a semicircle of radius \( r \) such that its area is maximized, we can follow these steps:

### Step 1: Understand the Geometry
We have a semicircle of radius \( r \) with its diameter along the x-axis. The rectangle will have its base on the diameter, and its top corners will touch the semicircle.

### Step 2: Define Variables
Let:
- \( x \) = half the width of the rectangle (from the center to one side)
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