Question

A cone is placed inside a cylinder. The apex of the cone touching the center of the top circle of the cylinder is highlighted with a dot. The cone with its base is drawn in dashed lines. The base of the cone is common with the base of the cylinder.

A sculptor is creating a design by carving 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 original cylinder?(1 point)
Responses

Answer 1

To find the volume of the original cylinder, we first need to understand the relationship between the volumes of the cone and the cylinder when they share the same radius and height.

The formulas for the volumes are as follows:

1. **Volume of a Cylinder**:
\[
V_{cylinder} = \pi r^2 h
\]

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

Since the cone and cylinder have the same radius \(r\) and height \(h\), we can express the volume of the remaining material (the cylinder minus the cone):

\[
V_{remaining} = V_{cylinder} - V_{cone}
\]

Substituting the volume formulas:

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

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

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

We know from the problem statement that the volume remaining is \(144 \text{ in}^3\):

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

To find the volume of the original cylinder \(V_{cylinder}\), we can solve for \(V_{cylinder}\):

\[
V_{cylinder} = \pi r^2 h = \frac{3}{2} V_{remaining}
\]
\[
V_{cylinder} = \frac{3}{2} \times 144 = 216 \text{ in}^3
\]

Therefore, the volume of the original cylinder is \(\boxed{216 \text{ in}^3}\).