To find the volume of cylinder B, we first need to determine the constant \( k \) using the information provided for cylinder A.
The volume formula is given by:
\[ V = kr^2h \]
For cylinder A, we have:
- Volume \( V = 197.82 \) cubic inches
- Radius \( r = 3 \) inches
- Height \( h = 7 \) inches
Substituting the values for cylinder A into the equation:
\[ 197.82 = k(3^2)(7) \]
Calculating \( 3^2 \) and substituting:
\[ 197.82 = k(9)(7) \]
\[ 197.82 = k(63) \]
Now, solving for \( k \):
\[ k = \frac{197.82}{63} \]
Calculating \( k \):
\[ k \approx 3.13905 \]
Now that we have \( k \), we can find the volume of cylinder B. Cylinder B has:
- Radius \( r = 4 \) inches
- Height \( h = 9 \) inches
Using the volume formula again:
\[ V_B = k r^2 h \]
Substituting the values:
\[ V_B = 3.13905 (4^2)(9) \]
Calculating \( 4^2 \):
\[ V_B = 3.13905 (16)(9) \]
Calculating \( 16 \times 9 \):
\[ V_B = 3.13905 (144) \]
Now multiplying \( k \) by \( 144 \):
\[ V_B \approx 3.13905 \times 144 \approx 451.22 \]
Thus, the volume of cylinder B is approximately:
\[ \boxed{451.22} \text{ cubic inches} \]