The volume of a cylinder varies jointly with the square of its radius and with its height: V=kr^2h

Cylinder A has a volume of 197.82
cubic inches and has a radius of 3
inches and a height of 7
inches. What is the volume of cylinder B, which has a radius of 4
inches and a height of 9
inches?

1 answer

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:

\[ \boxed{451.22} \text{ cubic inches} \]