This four-leaf clover consists of four coplanar circles. Each circle is externally tangent to two others, as shown. The two smaller circles are congruent, and the two larger circles are congruent. A square is constructed such that each of its four vertices is also the center of one of the four circles. Suppose the area of the square is 6 1/4 in2, and the combined area of the four circular leaves of the clover is 6 1/2ð in2. What is the positive difference between the radius of a large circular leaf and the radius of a small circular leaf? Express your answer as a common fraction.

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To find the positive difference between the radius of a large circular leaf and the radius of a small circular leaf, we can start by calculating the side length of the square.

We are given that the area of the square is 6 1/4 in^2. Recall that the area of a square is given by the formula A = s^2, where s is the side length of the square.

So, we can solve the equation 6 1/4 = s^2 to find the side length of the square.

First, let's convert the mixed number 6 1/4 to an improper fraction: 6 1/4 = 25/4.

The equation now becomes 25/4 = s^2.

To solve for s, we can take the square root of both sides of the equation.

√(25/4) = s.

The square root of 25 is 5, and the square root of 4 is 2.

So, s = 5/2.

This means the side length of the square is 5/2 inches.

Next, let's calculate the area of the square in terms of the side length.

The area of the square is given by A = s^2.

Substituting s = 5/2, we have A = (5/2)^2 = 25/4 = 6 1/4 in^2. This matches the given area.

Now, let's find the area of the circular leaves.

We are given that the combined area of the four circular leaves is 6 1/2π in^2.

Since the area of a circle is given by the formula A = πr^2, where r is the radius of the circle, we can set up an equation to find the combined area of the circular leaves.

Let's assume the radius of the small circular leaf is r, in inches.

So, the area of the small circular leaf is πr^2.

Since there are two small circular leaves and two large circular leaves, the combined area of the four circular leaves is 2(πr^2) + 2(πR^2), where R is the radius of the large circular leaf.

We are given that the combined area of the circular leaves is 6 1/2π in^2.

Therefore, we can set up the equation 2(πr^2) + 2(πR^2) = 6 1/2π.

Dividing both sides of the equation by 2π, we have r^2 + R^2 = 6 1/4.

Substituting the known area of the square (6 1/4) for R^2, we can rewrite the equation as r^2 + (5/2)^2 = 6 1/4.

r^2 + 25/4 = 6 1/4.

To simplify the equation, let's convert the mixed number 6 1/4 to an improper fraction: 6 1/4 = 25/4.

The equation becomes r^2 + 25/4 = 25/4.

Now, we can isolate r^2 by subtracting 25/4 from both sides of the equation:

r^2 = 25/4 - 25/4.

r^2 = 0.

This means that the radius of the small circular leaf is 0 inches.

However, this result does not make sense geometrically. A clover with four coplanar circles, as described in the problem, would not have a circle with a radius of 0.

Therefore, there seems to be an error in the problem statement or calculations.

Please double-check the given information or provide any additional details to proceed with finding the positive difference between the radii of the large and small circular leaves.