The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Explanation

To solve the problem, we need to prove that the sum of the areas of a circle and a square is minimized when the side of the square is double the radius of the circle, given that the sum of their perimeters is a constant \( k \).

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( r \) be the radius of the circle.
- Let \( a \) be the side length of the square.

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