Question
If the cylinder and the cone shown the same volume, then the radius and height of the cone could be which of these? Diameter 10 and height is 4: (radius of the cone = 5, height of the cone = 12; radius of the cone = 5, height of the cone = 4; radius of the cone = 15, height of the cone = 4; radius of the cone = 15, height of the cone = 12)
Answers
Answered by
GPT-4o mini
To find out which set of dimensions for the cone gives it the same volume as the cylinder, we need to use the formulas for the volume of a cylinder and a cone.
1. **Volume of a Cylinder**:
\[
V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}
\]
2. **Volume of a Cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}
\]
Given that the diameter of the cylinder is 10, the radius \( r_{\text{cylinder}} \) is \( \frac{10}{2} = 5 \).
Let’s consider \( h_{\text{cylinder}} \) to determine the volume of the cylinder. We can compute a general expression for volume in relation to the height of the cylinder.
3. **Volume of the Cylinder (assuming height = 4)**:
\[
V_{\text{cylinder}} = \pi (5)^2 (4) = 100\pi
\]
Now, we need to check which of the cone dimensions provided would result in a cone volume of \( 100\pi \).
### Options for the cone:
1. **Radius = 5, Height = 12**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (5^2)(12) = \frac{1}{3} \pi (25)(12) = \frac{300}{3} \pi = 100\pi
\]
2. **Radius = 5, Height = 4**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (5^2)(4) = \frac{1}{3} \pi (25)(4) = \frac{100}{3} \pi \neq 100\pi
\]
3. **Radius = 15, Height = 4**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (15^2)(4) = \frac{1}{3} \pi (225)(4) = \frac{900}{3} \pi = 300\pi \neq 100\pi
\]
4. **Radius = 15, Height = 12**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (15^2)(12) = \frac{1}{3} \pi (225)(12) = \frac{2700}{3} \pi = 900\pi \neq 100\pi
\]
### Conclusion:
The only dimensions for the cone that equal the volume of the cylinder when the cylinder has a diameter of 10 and height of 4 is:
- **Radius = 5, Height = 12**.
1. **Volume of a Cylinder**:
\[
V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}
\]
2. **Volume of a Cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}
\]
Given that the diameter of the cylinder is 10, the radius \( r_{\text{cylinder}} \) is \( \frac{10}{2} = 5 \).
Let’s consider \( h_{\text{cylinder}} \) to determine the volume of the cylinder. We can compute a general expression for volume in relation to the height of the cylinder.
3. **Volume of the Cylinder (assuming height = 4)**:
\[
V_{\text{cylinder}} = \pi (5)^2 (4) = 100\pi
\]
Now, we need to check which of the cone dimensions provided would result in a cone volume of \( 100\pi \).
### Options for the cone:
1. **Radius = 5, Height = 12**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (5^2)(12) = \frac{1}{3} \pi (25)(12) = \frac{300}{3} \pi = 100\pi
\]
2. **Radius = 5, Height = 4**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (5^2)(4) = \frac{1}{3} \pi (25)(4) = \frac{100}{3} \pi \neq 100\pi
\]
3. **Radius = 15, Height = 4**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (15^2)(4) = \frac{1}{3} \pi (225)(4) = \frac{900}{3} \pi = 300\pi \neq 100\pi
\]
4. **Radius = 15, Height = 12**:
\[
V_{\text{cone}} = \frac{1}{3} \pi (15^2)(12) = \frac{1}{3} \pi (225)(12) = \frac{2700}{3} \pi = 900\pi \neq 100\pi
\]
### Conclusion:
The only dimensions for the cone that equal the volume of the cylinder when the cylinder has a diameter of 10 and height of 4 is:
- **Radius = 5, Height = 12**.