Final answer:
The ratio of the perimeter of the square to the square of the diameter of the circle is π / 6.
Step-by-step explanation:
The given problem states that a circle has radius r and a rectangle has length x and width y. The area of the circle is twice the area of the rectangle. Since the rectangle is a square with an area of 9, we can find the side of the square and its perimeter. The side length of the square, s, is the square root of the area, which is 3 units (since 3×3=9). Therefore, the perimeter of the square is 4×3=12 units. The area of the circle is given as twice the area of the square, so Acircle = 2 × 9 = 18 units2. The formula for the area of a circle is A = πr2. By setting πr2 equal to 18, we can solve for the radius r. To find the square of the diameter of the circle, we will use the formula (2×r)2 = 4×r2. Now we calculate the values: r = √(18/π) and the diameter squared is (2×r)2 = 4×(18/π) = 18×(4/π). The final step is to compute the ratio of the perimeter of the square divided by the square of the diameter of the circle: Perimeter of square / (Square of diameter of circle) = 12 / (18 × (4/π)) = 12π / 72 = π / 6.