The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.

Answer

To find the curved surface area (CSA) of a frustum of a cone, we can use the formula:

\[ \text{CSA} = \pi (r_1 + r_2) l \]

where \( r_1 \) and \( r_2 \) are the radii of the two circular ends of the frustum, and \( l \) is the slant height.

### Step 1: Find the radii of the circular ends
The perimeters (circumferences) of the circular ends are given as 18 cm and 6 cm. We can find the radii using the formula for circumference:

\[ C = 2\pi r \]

For the first circular end (perimeter = 18 cm):

\[ 18 = 2\pi r_1 \implies r_1 = \frac{18}{2\pi} = \frac{9}{\pi} \text{ cm} \]

For the second circular end (perimeter = 6 cm):

\[ 6 = 2\pi r_2 \implies r_2 = \frac{6}{2\pi} = \frac{3}{\pi} \text{ cm} \]

### Step 2: Substitute the values into the CSA formula
Now that we have both radii and the slant height \( l = 4 \) cm, we can substitute these values into the CSA formula:

\[ \text{CSA} = \pi \left( \frac{9}{\pi} + \frac{3}{\pi} \right) \cdot 4 \]

### Step 3: Simplify the expression
Combine the terms inside the parentheses:

\[ \text{CSA} = \pi \left( \frac{9 + 3}{\pi} \right) \cdot 4 = \pi \left( \frac{12}{\pi} \right) \cdot 4 \]

Now, simplify:

\[ \text{CSA} = 12 \cdot 4 = 48 \text{ cm}^2 \]

### Final Answer
The curved surface area of the frustum is \( 48 \text{ cm}^2 \).

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