To determine the area of a circle with a radius of 6.2 inches, we start with the formula for the area of a circle:
\[ A = k r^2 \]
where \( A \) is the area, \( r \) is the radius, and \( k \) is the constant of variation.
From the problem, we know that a circle with a radius of 2 inches has an area of 12.568 square inches. We can use this information to find \( k \):
\[ 12.568 = k (2^2) \]
Calculating \( 2^2 \):
\[ 2^2 = 4 \]
Now we can plug this into our equation:
\[ 12.568 = k \times 4 \]
To find \( k \), we divide both sides by 4:
\[ k = \frac{12.568}{4} = 3.142 \]
Now that we have the value of \( k \), we can find the area of the circle with a radius of 6.2 inches:
\[ A = k (6.2^2) \]
Calculating \( 6.2^2 \):
\[ 6.2^2 = 38.44 \]
Now substitute \( k \) and \( 6.2^2 \) back into the area formula:
\[ A = 3.142 \times 38.44 \]
Calculating the product:
\[ A \approx 3.142 \times 38.44 \approx 120.634 \]
Thus, the area of the circle with a radius of 6.2 inches is approximately:
\[ \boxed{120.634} \text{ square inches.} \]