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

Question

54
in
.
3
, what is the volume of the amount remaining?(1 point)
Responses

40 in.3
40
in.
3
40 in. cubed

18 in.3
18
in.
3
18 in. cubed

36 in.3
36
in.
3
36 in. cubed

27 in.3
27
in.
3
27 in. cubed
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Answer 1

To find the volume of the remaining material after the cone is carved out from the cylinder, we first need to calculate the volume of the cone and then subtract that from the volume of the cylinder.

1. Volume of the cylinder (V_cylinder) is given as:
   \[ V_{\text{cylinder}} = 54 , \text{in}^3 \]

2. Volume of the cone (V_cone) is calculated using the formula:
   \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Since the cone and cylinder share the same radius (r) and height (h), let's denote the height and radius of the cylinder (and cone) simply as h and r.

   We know the volume of the cylinder:
   \[ V_{\text{cylinder}} = \pi r^2 h \] Given \( V_{\text{cylinder}} = 54 \):
   \[ \pi r^2 h = 54 \]

   To express the volume of the cone:
   \[ V_{\text{cone}} = \frac{1}{3} \times \pi r^2 h \] Substitute for \( \pi r^2 h \) from the cylinder:
   \[ V_{\text{cone}} = \frac{1}{3} \times 54 = 18 , \text{in}^3 \]

3. Now, subtract the volume of the cone from the volume of the cylinder to find the remaining volume:
   \[ \text{Remaining volume} = V_{\text{cylinder}} - V_{\text{cone}} = 54 - 18 = 36 , \text{in}^3 \]

Thus, the volume of the amount remaining after the cone is removed is 36 in³.