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 analyze the volume relationships among a sphere, a cylinder, and a cone that share the same radius \( r \), and where the cylinder and cone share the same height \( h \), we can start by writing the volume formulas for each shape:

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

Now let's explore the relationships given in the options:

1. **sphere = cylinder + cone**:
\[
V_s = V_c + V_{cone}
\]
\[
\frac{4}{3} \pi r^3 \neq \pi r^2 h + \frac{1}{3} \pi r^2 h
\]
This is false, as the left side represents the volume of the sphere while the right side does not equal it.

2. **3 cones = 1 cylinder**:
\[
3 V_{cone} = V_c
\]
\[
3 \left(\frac{1}{3} \pi r^2 h\right) = \pi r^2 h
\]
This is true, confirming that the volume of three cones is equal to the volume of one cylinder.

3. **1 cylinder = 2 cones + 1 sphere**:
\[
V_c = 2 V_{cone} + V_s
\]
\[
\pi r^2 h \neq 2 \left(\frac{1}{3} \pi r^2 h\right) + \frac{4}{3} \pi r^3
\]
This is false.

4. **sphere = cone = cylinder**:
This suggests all three shapes have the same volume, which is not true based on our volume formulas.

Thus, reviewing these options, the correct statement is **3 cones = 1 cylinder**.