Question

A 3-dimensional cone is placed on top of a 3-dimensional cylinder. One of the cylinder bases is common to the cone, with the same circumference.

A 3D modeler is designing a component of a castle in a video game. She places a cone on top of a cylinder so they line up and share the same radius. The cone and the cylinder share the same height. If the volume of the cone is 50 cm3
, what is the volume of the whole object?

Answer 1

To find the volume of the whole object, which includes both the cone and the cylinder, we need to calculate the volume of the cylinder and then add it to the volume of the cone.

1. **Volume of the Cone**: The volume of a cone is given by the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
We know from the problem that the volume of the cone is \(50 \, \text{cm}^3\).

2. **Height and Radius of the Cone**: Let the radius of the cone (and cylinder) be \(r\), and the height (which is the same for both the cone and cylinder) be \(h\). We can use the volume of the cone to express \(h\):
\[
50 = \frac{1}{3} \pi r^2 h
\]
Rearranging this gives:
\[
h = \frac{150}{\pi r^2}
\]

3. **Volume of the Cylinder**: The volume of a cylinder is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
We can substitute \(h\) into this equation:
\[
V_{\text{cylinder}} = \pi r^2 \left(\frac{150}{\pi r^2}\right)
\]
Simplifying:
\[
V_{\text{cylinder}} = 150 \, \text{cm}^3
\]

4. **Total Volume of the Whole Object**: Finally, we find the total volume by adding the volume of the cone and the volume of the cylinder:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 50 \text{ cm}^3 + 150 \text{ cm}^3
\]
\[
V_{\text{total}} = 200 \text{ cm}^3
\]

Thus, the volume of the whole object is \( \boxed{200 \text{ cm}^3} \).