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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

18 in.3
18 in. cubed

40 in.3
40 in. cubed

27 in.3
27 in. cubed

36 in.3
36 in. cubed

Answer 1

To find the volume of the remaining material after the cone has been carved out of the cylinder, we first need to calculate the volume of the cone.

1. The volume of a cylinder is given by the formula:

\[ V_{cylinder} = \pi r^2 h \]

2. The volume of a cone is given by the formula:

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

Since we know the volume of the cylinder is 54 in³, we can express this as:

\[ V_{cylinder} = 54 , \text{in}^3 \]

Since the cone shares the same radius and height as the cylinder, we can set up the equation for the volume of the cone.

3. To find the volume of the cone, we use the relationship:

\[ V_{cone} = \frac{1}{3} V_{cylinder} \]

So,

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

4. The volume of the remaining material in the cylinder after the cone has been carved out is:

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

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

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