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

cylinder = sphere – cone
cylinder = sphere – cone

cone = cylinder – sphere
cone = cylinder – sphere

cone = sphere – cylinder
cone = sphere – cylinder

Answer 1

To compare the volumes of the sphere, cylinder, and cone, we first need to write out their volume formulas based on a shared radius \( r \) and the height of the cylinder and cone as \( h = 2r \).

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

2. **Volume of the Cylinder (V_c)**:
\[
V_c = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3
\]

3. **Volume of the Cone (V_co)**:
\[
V_co = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Next, let's express the total volume of the cylinder and cone:
\[
V_c + V_co = 2 \pi r^3 + \frac{2}{3} \pi r^3
\]

To combine these, we need a common denominator:
\[
V_c + V_co = 2 \pi r^3 + \left(\frac{2}{3} \pi r^3\right) = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]

Now, we compare \( V_s \) to \( V_c + V_co \):
\[
V_s = \frac{4}{3} \pi r^3
\]
\[
V_c + V_co = \frac{8}{3} \pi r^3
\]

From this, it becomes clear that:
\[
V_c + V_co = 2 V_s
\]

Thus, rearranging gives us:
\[
V_s = \frac{1}{2} (V_c + V_co)
\]

This shows that:
\[
V_s < V_c + V_co
\]

Now, we can analyze the options:

1. **sphere = cylinder + cone** ⇒ \( \frac{4}{3} \pi r^3 \neq 2 \pi r^3 + \frac{2}{3} \pi r^3 \)

2. **cylinder = sphere – cone** ⇒ \( 2 \pi r^3 \neq \left(\frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3\right) \)

3. **cone = cylinder – sphere** ⇒ \( \frac{2}{3} \pi r^3 \neq \left(2 \pi r^3 - \frac{4}{3} \pi r^3\right) \)

4. **cone = sphere – cylinder** ⇒ \( \frac{2}{3} \pi r^3 \neq \left(\frac{4}{3} \pi r^3 - 2 \pi r^3\right) \)

None of the options provided correctly express the relationship we found.

Therefore, the correct conclusion based on the volume comparison is:
**$V_s$ is less than the sum of the volumes of the cylinder and the cone.** However, this is not directly reflected in the options given.