Given that the sphere, cylinder, and cone share the same radius \( r \), and the cylinder and cone have the same height \( h = 2r \), we can determine the relationships between their volumes by using their volume formulas.
1. **Volume of the Sphere:**
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
2. **Volume of the Cylinder:**
\[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3 \]
3. **Volume of the Cone:**
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, let's analyze the given statements with these results:
1. **Sphere - Cone = Cylinder:**
\[ \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = 2 \pi r^3 \]
\[ \frac{2}{3} \pi r^3 \ne 2 \pi r^3 \]
This statement is false.
2. **3 Cones = 1 Cylinder:**
\[ 3 \times \frac{2}{3} \pi r^3 = 2 \pi r^3 \]
\[ 2 \pi r^3 = 2 \pi r^3 \]
This statement is true.
3. **Sphere = Cylinder + Cone:**
\[ \frac{4}{3} \pi r^3 = 2 \pi r^3 + \frac{2}{3} \pi r^3 \]
\[ \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]
This statement is false.
4. **1 Cylinder = 2 Cones + 1 Sphere:**
\[ 2 \pi r^3 = 2 \times \frac{2}{3} \pi r^3 + \frac{4}{3} \pi r^3 \]
\[ 2 \pi r^3 = \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]
\[ 2 \pi r^3 \ne \frac{8}{3} \pi r^3 \]
This statement is false.
Therefore, the correct statement is:
\[
3 \text{ cones} = 1 \text{ cylinder}
\]
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
sphere - cone = cylinder
sphere - cone = cylinder
3 cones = 1 cylinder
3 cones = 1 cylinder
sphere = cylinder + cone
sphere = cylinder + cone
1 cylinder = 2 cones + 1 sphere
1 answer