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?

(1 point)

Answer 1

To find the total volume of the object made of a cone on top of a cylinder, we first need to find the volume of the cylinder and then add it to the volume of the cone.

Given:

- Volume of the cone \( V_{cone} = \frac{1}{3} \pi r^2 h \)
- Volume of the cylinder \( V_{cylinder} = \pi r^2 h \)

According to the information provided:
- The volume of the cone is \( 50 \, \text{cm}^3 \).
- The cone and the cylinder have the same radius \( r \) and height \( h \).

Let’s denote the volume of the cone as:
\[
V_{cone} = 50 \, \text{cm}^3
\]

The volume of the cylinder can be expressed in terms of the volume of the cone as follows:
\[
V_{cylinder} = 3 V_{cone}
\]
because the volume of the cone is a third of the volume of the cylinder when they have the same radius and height.

Thus, we can calculate the volume of the cylinder:
\[
V_{cylinder} = 3 \times 50 \, \text{cm}^3 = 150 \, \text{cm}^3
\]

Now, we find the total volume of the whole object (cone + cylinder):
\[
V_{total} = V_{cone} + V_{cylinder} = 50 \, \text{cm}^3 + 150 \, \text{cm}^3 = 200 \, \text{cm}^3
\]

Therefore, the volume of the whole object is:
\[
\boxed{200 \, \text{cm}^3}
\]