Question

Cones, Cylinders, & Spheres Online Practice
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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)
Responses

2 cones = 1 cylinder
2 cones = 1 cylinder

2 spheres = 1 cone
2 spheres = 1 cone

2 cones = 1 sphere
2 cones = 1 sphere

2 cylinders = 1 cone

Answer 1

To determine the relationship between the volumes of a sphere, cylinder, and cone with the given conditions, we can use the formulas for the volumes of these shapes:

1. **Sphere**: The volume \( V \) is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]

2. **Cylinder**: The volume \( V \) is given by the formula:
\[
V = \pi r^2 h
\]
Here, since the height \( h \) of the cylinder is twice the radius \( r \), we can substitute \( h \):
\[
V = \pi r^2 (2r) = 2 \pi r^3
\]

3. **Cone**: The volume \( V \) is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Again, substituting \( h = 2r \):
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now we have:

- **Sphere Volume**: \( \frac{4}{3} \pi r^3 \)
- **Cylinder Volume**: \( 2 \pi r^3 \)
- **Cone Volume**: \( \frac{2}{3} \pi r^3 \)

Now let's compare these volumes to find the relationships:

**Finding the relationships:**

1. **Comparing Cylinder to Cone:**
\[
\text{Cylinder volume} = 2 \pi r^3
\]
\[
\text{Cone volume} = \frac{2}{3} \pi r^3
\]
So,
\[
2 \text{ cones} = 2 \cdot \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 = \text{Sphere volume}
\]
Thus, \( 2 \text{ cones} = 1 \text{ sphere} \).

2. **Comparing 2 Cones to Cylinder:**
\[
2 \text{ cones} = 2 \cdot \left(\frac{2}{3} \pi r^3\right) = \frac{4}{3} \pi r^3 = \text{ Sphere volume}, \text{ so this does not hold.}
\]

Now substituting the findings into the options given:

- **2 cones = 1 cylinder**: False
- **2 spheres = 1 cone**: False
- **2 cones = 1 sphere**: True
- **2 cylinders = 1 cone**: False

Therefore, the correct answer is:
**2 cones = 1 sphere**.