A cylinder and a cone have the same height and the same radius. The volume of the cylinder is `24cm^(3)`. What will be the volume of the cone?

Answer

To find the volume of the cone given that the cylinder and cone have the same height and radius, and the volume of the cylinder is 24 cm³, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formulas**:
- The volume of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
- The volume of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

2. **Given Information**:
- We know that the volume of the cylinder is 24 cm³:
\[ V_{\text{cylinder}} = 24 \, \text{cm}^3 \]
- Therefore, we can write:
\[ \pi r^2 h = 24 \quad \text{(Equation 1)} \]

3. **Relate the Volume of the Cone to the Cylinder**:
- The volume of the cone can be expressed using the volume of the cylinder:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

4. **Substitute the Volume of the Cylinder into the Cone's Volume Formula**:
- From Equation 1, we know that \(\pi r^2 h = 24\). We can substitute this into the volume formula for the cone:
\[ V_{\text{cone}} = \frac{1}{3} \times 24 \]

5. **Calculate the Volume of the Cone**:
- Now, calculate:
\[ V_{\text{cone}} = \frac{24}{3} = 8 \, \text{cm}^3 \]

6. **Final Answer**:
- Therefore, the volume of the cone is:
\[ V_{\text{cone}} = 8 \, \text{cm}^3 \]