The area \( A \) of a circle is given by the formula:
\[ A = k r^2 \]
where \( k \) is the constant of proportionality, and \( r \) is the radius of the circle.
From the information provided, we know that for a circle with a radius of 2 inches, the area is 12.568 square inches. We can use this information to find \( k \).
Substituting the values we have:
\[ 12.568 = k \cdot (2^2) \]
\[ 12.568 = k \cdot 4 \]
Solving for \( k \):
\[ k = \frac{12.568}{4} = 3.142 \]
Now that we have the value of \( k \), we can find the area of a circle with a radius of 5.7 inches.
Using the formula again:
\[ A = k r^2 \]
Substituting \( k = 3.142 \) and \( r = 5.7 \):
\[ A = 3.142 \cdot (5.7^2) \]
Calculating \( 5.7^2 \):
\[ 5.7^2 = 32.49 \]
Then substituting back into the area formula:
\[ A = 3.142 \cdot 32.49 \approx 102.12658 \]
Thus, the area of a circle with a radius of 5.7 inches is approximately \( 102.13 \) square inches.