Question

Volume of Cones, Cylinders, and Spheres Quick Check
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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?(1 point)
Responses

cylinder = sphere – cone
cylinder = sphere – cone

cone = sphere – cylinder
cone = sphere – cylinder

sphere = cylinder + cone
sphere = cylinder + cone

cone = cylinder – sphere

Answer 1

Let's denote the radius as \( r \) and the height of the cylinder and cone as \( h = 2r \).

The volume formulas for each shape are as follows:

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

- **Cylinder**:
\[
V_{cylinder} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]

- **Cone**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now, let's compare the volumes.

1. **Volume of the Cylinder**:
\[
V_{cylinder} = 2\pi r^3
\]

2. **Volume of the Cone**:
\[
V_{cone} = \frac{2}{3} \pi r^3
\]

Now we can check the relationship between them:

### Option checking:

1. **cylinder = sphere – cone**:
\[
2\pi r^3 \stackrel{?}{=} \frac{4}{3}\pi r^3 - \frac{2}{3}\pi r^3
\]
\[
2\pi r^3 = \frac{2}{3}\pi r^3 \quad \text{(False)}
\]

2. **cone = sphere – cylinder**:
\[
\frac{2}{3}\pi r^3 \stackrel{?}{=} \frac{4}{3}\pi r^3 - 2\pi r^3
\]
\[
\frac{2}{3}\pi r^3 = \frac{4}{3}\pi r^3 - \frac{6}{3}\pi r^3
\]
\[
\frac{2}{3}\pi r^3 = -\frac{2}{3}\pi r^3 \quad \text{(False)}
\]

3. **sphere = cylinder + cone**:
\[
\frac{4}{3}\pi r^3 \stackrel{?}{=} 2\pi r^3 + \frac{2}{3}\pi r^3
\]
\[
\frac{4}{3}\pi r^3 = \left(\frac{6}{3}\pi r^3 + \frac{2}{3}\pi r^3\right) \quad \text{(True)}
\]

4. **cone = cylinder – sphere**:
\[
\frac{2}{3}\pi r^3 \stackrel{?}{=} 2\pi r^3 - \frac{4}{3}\pi r^3
\]
\[
\frac{2}{3}\pi r^3 = \left(\frac{6}{3}\pi r^3 - \frac{4}{3}\pi r^3\right) \quad \text{(False)}
\]

### Conclusion:
The true relationship is:
\[
\text{sphere} = \text{cylinder} + \text{cone}
\]

So the correct response is:
**sphere = cylinder + cone**