The perimeters of the two circualr ends of a frustum of a cone are 48 cm and 36 cm. If the height of the frustum is 11 cm, find its volume and curved surface area.

Explanation

To solve the problem of finding the volume and curved surface area of a frustum of a cone given the perimeters of its circular ends and its height, we can follow these steps:

### Step 1: Understand the given information
- The perimeter (circumference) of the upper circular end (C1) = 48 cm
- The perimeter (circumference) of the lower circular end (C2) = 36 cm
- Height (h) of the frustum = 11 cm

### Step 2: Calculate the radii of the circular ends
The formula for the circumference of a circle is given by:
\[ C = 2\pi r \]
Where \( r \) is the radius.

For the upper end:
\[ C1 = 2\pi r_1 \]
\[ 48 = 2\pi r_1 \]
\[ r_1 = \frac{48}{2\pi} = \frac{24}{\pi} \]

For the lower end:
\[ C2 = 2\pi r_2 \]
\[ 36 = 2\pi r_2 \]
\[ r_2 = \frac{36}{2\pi} = \frac{18}{\pi} \]

### Step 3: Calculate the slant height (l)
The slant height \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{(r_1 - r_2)^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{\left(\frac{24}{\pi} - \frac{18}{\pi}\right)^2 + 11^2} \]
\[ = \sqrt{\left(\frac{6}{\pi}\right)^2 + 121} \]
\[ = \sqrt{\frac{36}{\pi^2} + 121} \]

### Step 4: Substitute the value of \(\pi\) and simplify
Using \(\pi \approx \frac{22}{7}\):
\[ l = \sqrt{\frac{36}{\left(\frac{22}{7}\right)^2} + 121} \]
\[ = \sqrt{\frac{36 \cdot 49}{484} + 121} \]
\[ = \sqrt{\frac{1764}{484} + 121} \]
\[ = \sqrt{\frac{1764 + 58484}{484}} \]
\[ = \sqrt{\frac{60248}{484}} \]
\[ = \sqrt{124.5} \approx 11.16 \text{ cm} \]

### Step 5: Calculate the volume of the frustum
The volume \( V \) of the frustum is given by:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]
Substituting the values:
\[ V = \frac{1}{3} \pi \cdot 11 \left(\left(\frac{24}{\pi}\right)^2 + \left(\frac{18}{\pi}\right)^2 + \left(\frac{24}{\pi}\right)\left(\frac{18}{\pi}\right)\right) \]
\[ = \frac{1}{3} \cdot \frac{22}{7} \cdot 11 \left(\frac{576}{\pi^2} + \frac{324}{\pi^2} + \frac{432}{\pi^2}\right) \]
\[ = \frac{1}{3} \cdot \frac{22}{7} \cdot 11 \cdot \frac{1332}{\pi^2} \]
\[ = \frac{22 \cdot 11 \cdot 1332}{21 \cdot 7} \]
\[ \approx 1554 \text{ cm}^3 \]

### Step 6: Calculate the curved surface area (CSA)
The formula for the curved surface area \( A \) of the frustum is:
\[ A = \pi (r_1 + r_2) l \]
Substituting the values:
\[ A = \pi \left(\frac{24}{\pi} + \frac{18}{\pi}\right) l \]
\[ = \pi \left(\frac{42}{\pi}\right) l \]
\[ = 42l \]
Substituting \( l \approx 11.16 \):
\[ A \approx 42 \cdot 11.16 \approx 468.91 \text{ cm}^2 \]

### Final Answers
- Volume of the frustum \( V \approx 1554 \text{ cm}^3 \)
- Curved Surface Area \( A \approx 468.91 \text{ cm}^2 \)