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

sphere = cylinder + cone
sphere = cylinder + cone

cone = cylinder – sphere
cone = cylinder – sphere

cylinder = sphere – cone
cylinder = sphere – cone

cone = sphere – cylinder
cone = sphere – cylinder

Answer 1

To analyze this problem, we first need to recall the formulas for the volumes of a sphere, cylinder, and cone.

1. **Volume of a sphere** (radius \( r \)):
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]

2. **Volume of a cylinder** (radius \( r \) and height \( h \)):
\[
V_{cylinder} = \pi r^2 h
\]
Given that \( h = 2r \):
\[
V_{cylinder} = \pi r^2 (2r) = 2 \pi r^3
\]

3. **Volume of a cone** (radius \( r \) and height \( h \)):
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Again, using \( h = 2r \):
\[
V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now, we can express the volumes for the sphere, cylinder, and cone:
- \( V_{sphere} = \frac{4}{3} \pi r^3 \)
- \( V_{cylinder} = 2 \pi r^3 \)
- \( V_{cone} = \frac{2}{3} \pi r^3 \)

Next, let's compare the volumes:

1. **Bundle the cone and cylinder**:
\[
V_{cylinder} - V_{cone} = 2 \pi r^3 - \frac{2}{3} \pi r^3
\]
To simplify this, let's express \( 2 \pi r^3 \) with a common denominator:
\[
2 \pi r^3 = \frac{6}{3} \pi r^3
\]
Now, substituting this back:
\[
V_{cylinder} - V_{cone} = \frac{6}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 = V_{sphere}
\]

Thus, we can conclude:
\[
V_{cylinder} - V_{cone} = V_{sphere}
\]

In summary, the correct relationship between the volumes is:
**cylinder = sphere + cone**

None of the provided options directly state that, but since default options appear to have stated in a different way, we have:
\[
\boxed{cylinder = sphere + cone}
\] (rephrased correctly)

Given the options provided in the original question:
- Sphere = Cylinder + Cone : **Correct!**
- Cone = Cylinder - Sphere: Incorrect
- Cylinder = Sphere - Cone: Incorrect
- Cone = Sphere - Cylinder: Incorrect

The first response corresponds to the correct relation.