The frustum of a right circular cone has the radii of base 4 cm, of the top 2 cm and a height of 6 cm. Find the volume of the frustum.

Options

A. `115 cm^(3) `
B. `156 cm^(3) `
C. `185 cm^(3) `
D. `176 cm^(3) `

Answer

To find the volume of the frustum of a right circular cone, we can use the formula:

\[ V = \frac{1}{3} \pi h \left( R^2 + r^2 + Rr \right) \]

where:
- \( V \) is the volume,
- \( h \) is the height of the frustum,
- \( R \) is the radius of the base,
- \( r \) is the radius of the top.

### Step-by-Step Solution:

1. **Identify the values**:
- Radius of the base, \( R = 4 \) cm
- Radius of the top, \( r = 2 \) cm
- Height of the frustum, \( h = 6 \) cm

2. **Substitute the values into the formula**:
\[ V = \frac{1}{3} \pi (6) \left( 4^2 + 2^2 + 4 \times 2 \right) \]

3. **Calculate \( R^2 \), \( r^2 \), and \( Rr \)**:
- \( R^2 = 4^2 = 16 \)
- \( r^2 = 2^2 = 4 \)
- \( Rr = 4 \times 2 = 8 \)

4. **Add these values together**:
\[ R^2 + r^2 + Rr = 16 + 4 + 8 = 28 \]

5. **Substitute back into the volume formula**:
\[ V = \frac{1}{3} \pi (6) (28) \]

6. **Calculate the volume**:
\[ V = \frac{1}{3} \times \pi \times 6 \times 28 \]
\[ V = \frac{1}{3} \times \frac{22}{7} \times 6 \times 28 \]

7. **Simplify the expression**:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 168 \]
(since \( 6 \times 28 = 168 \))

8. **Calculate \( \frac{168}{3} \)**:
\[ \frac{168}{3} = 56 \]

9. **Final calculation**:
\[ V = \frac{22}{7} \times 56 = \frac{1232}{7} \approx 176 \text{ cm}^3 \]

Thus, the volume of the frustum is \( 176 \text{ cm}^3 \).