Question

a sculptor is creating a design by carvinv a cone out of a cylinder. The cone and cylinder share the same radius and height. if the volume remaining is 144 in. ^3, what was the volume of the origional cylinnder?

Answer 1

To solve this problem, we need to use the formulas for the volume of a cylinder and a cone.

1. **Volume of a Cylinder (\( V_c \))**:
\[
V_c = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

2. **Volume of a Cone (\( V_{cone} \))**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]

Given that the cone is carved out of the cylinder and they share the same radius \( r \) and height \( h \), we can express the remaining volume after carving the cone out of the cylinder as follows:

\[
\text{Volume remaining} = V_c - V_{cone}
\]

Substituting in the formulas for the volumes:

\[
\text{Volume remaining} = \pi r^2 h - \frac{1}{3} \pi r^2 h
\]

Factoring out \( \pi r^2 h \):

\[
\text{Volume remaining} = \pi r^2 h \left(1 - \frac{1}{3}\right) = \pi r^2 h \left(\frac{2}{3}\right)
\]

Given that the volume remaining is \( 144 \, \text{in}^3 \):

\[
\frac{2}{3} \pi r^2 h = 144
\]

To find the original volume of the cylinder:

\[
V_c = \pi r^2 h = \frac{3}{2} \times 144 = 216 \, \text{in}^3
\]

Thus, the volume of the original cylinder is:

\[
\boxed{216 \, \text{in}^3}
\]