Question

Suppose a sphere, cylinder, and cone all share the same radius, and
the cylinder and cone also share the same height.
Which of the following is true about the relationship between the
volume formulas for the sphere, cylinder, and cone?
(1 point)
sphere = cylinder + cone
3 cones = 1 cylinder
1 cylinder = 2 cones + 1 sphere
sphere - cone = cylinder

Answer 1

To compare the volumes of the sphere, cylinder, and cone, we first need to know their volume formulas.

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

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

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

Here, \(r\) is the radius and \(h\) is the height. Given that the cylinder and cone share the same height, we will use the same \(h\) for both.

Now, let's evaluate the provided relationships one by one.

1. **sphere = cylinder + cone**:
\[
V_s \neq V_c + V_{cone}
\]
Because \(V_c = \pi r^2 h\) and \(V_{cone} = \frac{1}{3} \pi r^2 h\), therefore:
\[
V_c + V_{cone} = \pi r^2 h + \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^2 h
\]
So this statement is **false**.

2. **3 cones = 1 cylinder**:
\[
3 \cdot V_{cone} = V_c
\]
Substituting the volumes gives:
\[
3 \cdot \frac{1}{3} \pi r^2 h = \pi r^2 h
\]
Thus, it is correct that \(3 \text{ cones} = 1 \text{ cylinder}\). So this statement is **true**.

3. **1 cylinder = 2 cones + 1 sphere**:
To check:
\[
V_c = 2 V_{cone} + V_s
\]
Substituting gives:
\[
\pi r^2 h \neq 2 \cdot \frac{1}{3} \pi r^2 h + \frac{4}{3} \pi r^3
\]
This is a complex evaluation and doesn't hold true in general. Thus, this statement is **false**.

4. **sphere - cone = cylinder**:
To check:
\[
V_s - V_{cone} = V_c
\]
Substituting gives:
\[
\frac{4}{3} \pi r^3 - \frac{1}{3} \pi r^2 h \neq \pi r^2 h
\]
This would not generally hold true, hence this statement is **false**.

Thus, the only true statement among the options is:

\[
\text{3 cones} = \text{1 cylinder}
\]