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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 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

192 in.3
192 in. cubed

180 in.3
180 in. cubed

216 in.3
216 in. cubed

288 in.3
288 in. cubed
Remaining Attempts : 3
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Answer 1

To find the volume of the original cylinder, you first need to understand the relationship between the volumes of the cylinder and the cone. The formula for the volume of a cylinder (V_cylinder) is given by:

$$V_{cylinder}=\pi r^{2}h$$

The formula for the volume of a cone (V_cone) is given by:

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

Since the cone and cylinder share the same radius (r) and height (h), the volume of the remaining material after carving the cone from the cylinder is:

$$V_{remaining}=V_{cylinder}-V_{cone}$$

Substituting the volumes into the equation gives:

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

Factoring out $\pi r^{2}h$:

$$V_{remaining}=\pi r^{2}h(1-\frac{1}{3})=\pi r^{2}h(\frac{3}{3}-\frac{1}{3})=\frac{2}{3}\pi r^{2}h$$

Now, if $V_{remaining}=144$ in³, then we can set up the equation:

$$\frac{2}{3}{V}_{cylinder}=144$$

To find the volume of the cylinder, solve for $V_{cylinder}$:

$$V_{cylinder}=\frac{144\cdot 3}{2}=216,{\text{in}}^3$$

Thus, the volume of the original cylinder was 216 in³.

So the correct response is:

216 in.3 (or 216 in. cubed)