Question

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

cone = sphere – cylinder
cone = sphere – cylinder

cylinder = sphere – cone
cylinder = sphere – cone

Answer 1

To answer this question, let's first write down the volume formulas for the sphere, cylinder, and cone based on the given dimensions.

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

2. **Cylinder**: The volume \( V_c \) of a cylinder with radius \( r \) and height \( h = 2r \) is given by:
\[
V_c = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3
\]

3. **Cone**: The volume \( V_{cone} \) of a cone with radius \( r \) and height \( h = 2r \) is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now we can compare the volumes:

- Volume of the sphere:
\[
V_s = \frac{4}{3} \pi r^3
\]

- Volume of the cylinder:
\[
V_c = 2 \pi r^3
\]

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

Next, we can see how these formulas compare to each other:

1. To check if \( V_s = V_c + V_{cone} \):
\[
V_c + V_{cone} = 2 \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]
This is not equal to \( V_s \).

2. To check if \( V_{cone} = V_c - V_s \):
\[
V_c - V_s = 2 \pi r^3 - \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 = V_{cone}
\]
This statement holds true.

3. To check if \( V_{cone} = V_s - V_c \):
\[
V_s - V_c = \frac{4}{3} \pi r^3 - 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 = -\frac{2}{3} \pi r^3
\]
This does not hold.

4. To check if \( V_c = V_s - V_{cone} \):
\[
V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
This does not hold.

Based on the calculations, the relationship that holds true is:

**cone = cylinder – sphere.**

So, the correct response is:
**cone = cylinder – sphere**.