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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

27 in.3

18 in.3

36 in.3

40 in.3

Answer 1

To find the volume of the remaining part of the cylinder after the cone has been carved out, we'll first need to calculate the volume of the cone and then subtract that from the volume of the cylinder.

The formula for the volume of a cylinder \( V_c \) is:

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

The volume of a cone \( V_{cone} \) is given by:

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

Since the problem states that the cylinder has a volume of 54 in³, we know:

\[ V_c = 54 \text{ in}^3 \]

We can express the volume of the cone in terms of the volume of the cylinder:

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

Now, to find the volume of the amount remaining in the cylinder after the cone is removed, we subtract the volume of the cone from the volume of the cylinder:

\[ V_{remaining} = V_c - V_{cone} = 54 \text{ in}^3 - 18 \text{ in}^3 = 36 \text{ in}^3 \]

Thus, the volume of the amount remaining is 36 in³.