Question

Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?
sphere = cylinder + cone
sphere = cylinder + cone

cone = cylinder – sphere
cone = cylinder – sphere

cone = sphere – cylinder
cone = sphere – cylinder

cylinder = sphere – cone

Answer 1

To analyze the relationships between the volumes of the sphere, cylinder, and cone, we can use their respective volume formulas:

1. **Sphere**: The volume \( V_{\text{sphere}} \) of a sphere is given by:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

2. **Cylinder**: The volume \( V_{\text{cylinder}} \) of a cylinder is given by:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Given that the height \( h \) of the cylinder is twice the radius \( r \) (i.e., \( h = 2r \)), we can substitute to find:
\[
V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3
\]

3. **Cone**: The volume \( V_{\text{cone}} \) of a cone is given by:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Similarly, since the height \( h \) of the cone is also \( 2r \):
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now we can compare the volumes we've calculated:

- Sphere volume:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

- Cylinder volume:
\[
V_{\text{cylinder}} = 2\pi r^3
\]

- Cone volume:
\[
V_{\text{cone}} = \frac{2}{3} \pi r^3
\]

Next, we can sum the volumes of the cylinder and the cone:
\[
V_{\text{cylinder}} + V_{\text{cone}} = 2\pi r^3 + \frac{2}{3} \pi r^3
\]
To combine these volumes, we convert \( 2\pi r^3 \) to a fraction with a denominator of 3:
\[
2\pi r^3 = \frac{6}{3} \pi r^3
\]
Adding them together:
\[
V_{\text{cylinder}} + V_{\text{cone}} = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]

Now we will compare \( \frac{8}{3} \pi r^3 \) to \( V_{\text{sphere}} \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
We see that \( V_{\text{cylinder}} + V_{\text{cone}} = \frac{8}{3} \pi r^3 \neq \frac{4}{3} \pi r^3 \), so the first relationship does not hold.

Next, let's explore if the cone \( = \text{cylinder} - \text{sphere} \):
To check this, we can calculate \( V_{\text{cylinder}} - V_{\text{sphere}} \):
\[
V_{\text{cylinder}} - V_{\text{sphere}} = 2\pi r^3 - \frac{4}{3} \pi r^3
\]
Converting \( 2\pi r^3 \) to a fraction with a denominator of 3:
\[
2\pi r^3 = \frac{6}{3} \pi r^3
\]
Then:
\[
V_{\text{cylinder}} - V_{\text{sphere}} = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 = V_{\text{cone}}
\]
This confirms that:
\[
V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}}
\]

Thus, we conclude that the correct relationship among the options given is:

**cone = cylinder – sphere**