The area \( A \) of a circle varies directly as the square of its radius \( r \). This can be expressed with the formula:
\[ A = k r^2 \]
where \( k \) is the constant of proportionality. Given that the area of a circle with radius 2 inches is 12.568 square inches, we can first find \( k \):
\[ 12.568 = k (2^2) \]
\[ 12.568 = k (4) \]
\[ k = \frac{12.568}{4} = 3.142 \]
Now that we have the value of \( k \), we can use it to find the area of a circle with a radius of 6.2 inches:
\[ A = k r^2 \] \[ A = 3.142 (6.2^2) \]
Calculating \( 6.2^2 \):
\[ 6.2^2 = 38.44 \]
Now, substituting back into the area formula:
\[ A = 3.142 \times 38.44 \]
Calculating this gives:
\[ A \approx 120.63408 \]
Rounding this to the nearest thousandth, the area of a circle with a radius of 6.2 inches is approximately:
\[ A \approx 120.634 \text{ in}^2 \]