Question

Use the image to answer the 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 created a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume of the cylinder before removing the cone is 54 in.3
, what is the volume of the amount remaining?(1 point)
Responses

18 in.3
18 in. cubed

27 in.3
27 in. cubed

40 in.3
40 in. cubed

36 in.3

Answer 1

To find the volume of the remaining amount after carving the cone out of the cylinder, we first need to determine the volume of the cone.

The volume \( V \) of a cylinder is given by the formula:

\[
V = \pi r^2 h
\]

The volume \( V \) of a cone is given by the formula:

\[
V = \frac{1}{3} \pi r^2 h
\]

Since the cone and the cylinder have the same radius \( r \) and height \( h \), we can express the volume of the cylinder and the cone in terms of \( r \) and \( h \).

Given that the volume of the cylinder is \( 54 \, \text{in}^3 \),

\[
\pi r^2 h = 54
\]

Now, we find the volume of the cone:

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

Substituting the expression for \( \pi r^2 h \):

\[
V_{cone} = \frac{1}{3} \times 54 = 18 \, \text{in}^3
\]

Now, we can find the remaining volume after removing the cone from the cylinder:

\[
\text{Remaining Volume} = \text{Volume of Cylinder} - \text{Volume of Cone}
\]
\[
\text{Remaining Volume} = 54 - 18 = 36 \, \text{in}^3
\]

So, the volume of the amount remaining is:

\[
\text{Remaining Volume} = 36 \, \text{in}^3
\]

Thus, the correct answer is **36 in.³**.