The perimeter of the ends of a frustum are 48 cm and 36 cm. If the height of the frustum is 11 cm, find its volume.

Answer

To find the volume of the frustum given the perimeters of its ends and its height, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Perimeter of the smaller end (P1) = 36 cm
- Perimeter of the larger end (P2) = 48 cm
- Height of the frustum (h) = 11 cm

2. **Calculate the radii of the ends using the perimeter formula**:
- The formula for the perimeter (circumference) of a circle is given by:
\[ P = 2\pi r \]
- For the smaller end:
\[ 36 = 2\pi r_1 \implies r_1 = \frac{36}{2\pi} = \frac{18}{\pi} \text{ cm} \]
- For the larger end:
\[ 48 = 2\pi r_2 \implies r_2 = \frac{48}{2\pi} = \frac{24}{\pi} \text{ cm} \]

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

4. **Substitute the values into the volume formula**:
- Substitute \( h = 11 \) cm, \( r_1 = \frac{18}{\pi} \) cm, and \( r_2 = \frac{24}{\pi} \) cm into the volume formula:
\[ V = \frac{1}{3} \pi \times 11 \left( \left(\frac{18}{\pi}\right)^2 + \left(\frac{24}{\pi}\right)^2 + \left(\frac{18}{\pi}\right) \left(\frac{24}{\pi}\right) \right) \]

5. **Calculate \( r_1^2 \), \( r_2^2 \), and \( r_1 r_2 \)**:
- \( r_1^2 = \left(\frac{18}{\pi}\right)^2 = \frac{324}{\pi^2} \)
- \( r_2^2 = \left(\frac{24}{\pi}\right)^2 = \frac{576}{\pi^2} \)
- \( r_1 r_2 = \left(\frac{18}{\pi}\right) \left(\frac{24}{\pi}\right) = \frac{432}{\pi^2} \)

6. **Combine the terms**:
- Now substitute these values back into the volume formula:
\[ V = \frac{1}{3} \pi \times 11 \left( \frac{324}{\pi^2} + \frac{576}{\pi^2} + \frac{432}{\pi^2} \right) \]
- Combine the fractions:
\[ V = \frac{1}{3} \pi \times 11 \left( \frac{1320}{\pi^2} \right) \]

7. **Simplify the expression**:
- Simplifying gives:
\[ V = \frac{1}{3} \times 11 \times \frac{1320}{\pi} = \frac{14520}{3\pi} = \frac{4840}{\pi} \text{ cm}^3 \]

8. **Approximate the volume using \( \pi \approx \frac{22}{7} \)**:
- Substitute \( \pi \):
\[ V \approx \frac{4840 \times 7}{22} = 1540 \text{ cm}^3 \]

### Final Answer:
The volume of the frustum is approximately **1540 cm³**.