Cones, Cylinders, & Spheres Online Practice

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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

180 in.3
180 in. cubed

192 in.3
192 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, we first need to use the information provided about the cone and the cylinder.

Since both the cone and the cylinder share the same radius (r) and height (h), we can use the volume formulas for both shapes:

1. Volume of a cylinder: \[ V_{cylinder} = \pi r^2 h \]

2. Volume of a cone: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]

The volume of the remaining part of the cylinder after carving out the cone is given as: \[ V_{remaining} = V_{cylinder} - V_{cone} \]

Substituting the formulas into this equation: \[ V_{remaining} = \pi r^2 h - \frac{1}{3} \pi r^2 h \]

Factoring out \(\pi r^2 h\): \[ V_{remaining} = \pi r^2 h \left(1 - \frac{1}{3}\right) = \pi r^2 h \left(\frac{2}{3}\right) \]

This means: \[ V_{remaining} = \frac{2}{3} \pi r^2 h \]

Given that the remaining volume is 144 in³, we set up the equation: \[ \frac{2}{3} \pi r^2 h = 144 \]

To find the volume of the original cylinder, we multiply both sides of the previous equation by \(\frac{3}{2}\) to isolate \(V_{cylinder}\): \[ V_{cylinder} = \frac{3}{2} \times 144 = 216 , \text{in}^3 \]

Thus, the volume of the original cylinder is: \[ \boxed{216 , \text{in}^3} \]