Question

Suppose a sphere, cylinder, and cone all share the same radius and 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)

3 cones = 1 cylinder 3 cones = 1 cylinder sphere = cylinder + cone sphere = cylinder + cone sphere - cone = cylinder sphere - cone = cylinder 1 cylinder = 2 cones + 1 sphere

Answer 1

To find the correct relationship between the volumes of a sphere, cylinder, and cone, we first need to calculate the volumes of each shape using their respective formulas.

1. **Volume of the Sphere (V_s)**:
\[
V_s = \frac{4}{3} \pi r^3
\]

2. **Volume of the Cylinder (V_c)**: The height (h) of the cylinder is twice the radius (r), so \( h = 2r \).
\[
V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]

3. **Volume of the Cone (V_cone)**: The height (h) of the cone is also \( 2r \) (same as the height of the cylinder).
\[
V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Next, we compare the volumes:

- **Three cones**:
\[
3 V_{cone} = 3 \left( \frac{2}{3} \pi r^3 \right) = 2 \pi r^3
\]

Since the volume of the cylinder is also \( 2 \pi r^3 \), we find:
\[
3 V_{cone} = V_c
\]
So, \( 3 \text{ cones} = 1 \text{ cylinder} \).

- **Comparing the sphere and the cylinder**:
\[
V_s = \frac{4}{3} \pi r^3 \quad \text{and} \quad V_c = 2\pi r^3
\]

To check if the sphere is equal to the cylinder plus the cone:
\[
V_c + V_{cone} = 2\pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \quad (not equal to V_s)
\]

Now let's check:
- **Sphere minus cone**:
\[
V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \quad (which is not equal to V_c)
\]

Given the calculations, we conclude that the correct relationship is:
\[
\boxed{3 \text{ cones} = 1 \text{ cylinder}}
\]

This statement is true and reflects the relationship accurately among the three volumes based on the given dimensions.