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 will follow these steps:

### Step 1: Understand the Geometry
We have a semicircle with radius \( r \) and a rectangle inscribed in it. The rectangle has its base on the diameter of the semicircle. Let the width of the rectangle be \( 2x \) (since it extends equally on both sides of the center) and the height be \( y \).

### Step 2: Relate the Dimensions to the Radius
Using the Pythagorean theorem for the right triangle formed by the radius, the height, and half the width, we have:
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