Find the volume of a frustum of a cone of height 14cm and the radii of its two circular ends are 4 cm and 2 cm.

Answer

To find the volume of a frustum of a cone, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Height (h) = 14 cm
- Radius of the larger circular end (r1) = 4 cm
- Radius of the smaller circular end (r2) = 2 cm

2. **Use the formula for the volume of a frustum of a cone**:
The formula for the volume \( V \) of a frustum of a cone is given by:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]

3. **Substitute the values into the formula**:
- Here, we will use \( \pi \approx \frac{22}{7} \).
- Substitute \( h = 14 \), \( r_1 = 4 \), and \( r_2 = 2 \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times 14 \times (4^2 + 2^2 + 4 \times 2) \]

4. **Calculate \( r_1^2 \), \( r_2^2 \), and \( r_1 r_2 \)**:
- \( r_1^2 = 4^2 = 16 \)
- \( r_2^2 = 2^2 = 4 \)
- \( r_1 r_2 = 4 \times 2 = 8 \)

5. **Sum these values**:
\[ r_1^2 + r_2^2 + r_1 r_2 = 16 + 4 + 8 = 28 \]

6. **Substitute back into the volume formula**:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 14 \times 28 \]

7. **Simplify the expression**:
- First, simplify \( \frac{14}{7} = 2 \):
\[ V = \frac{1}{3} \times 22 \times 2 \times 28 \]
- Now calculate \( 22 \times 2 = 44 \):
\[ V = \frac{1}{3} \times 44 \times 28 \]

8. **Calculate \( 44 \times 28 \)**:
\[ 44 \times 28 = 1232 \]

9. **Divide by 3**:
\[ V = \frac{1232}{3} \approx 410.67 \]

10. **State the final answer with the appropriate unit**:
The volume of the frustum of the cone is approximately \( 410.67 \, \text{cm}^3 \).