Question

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

Hint:

Whenever a problem gives circumferences instead of radii for a frustum, using the formula \(CSA = \frac{1}{2}(C_1 + C_2)l\) is much faster than calculating the radii first and then using the standard formula.

Answer 1

Step 1: Understanding the Concept:
A frustum of a cone is the portion of a cone left after its top has been cut off by a plane parallel to the base. We need to find its curved surface area (CSA) using the given slant height and the circumferences of its circular top and bottom bases.

Step 2: Key Formula or Approach:
The standard formula for the Curved Surface Area of a frustum is \(CSA = \pi(r_1 + r_2)l\), where \(r_1\) and \(r_2\) are the radii of the two circular ends and \(l\) is the slant height. A more direct formula using the circumferences (\(C_1\) and \(C_2\)) can be derived and used. Since \(C = 2\pi r\), we have \(r = C/(2\pi)\). So, \(r_1+r_2 = \frac{C_1}{2\pi} + \frac{C_2}{2\pi} = \frac{C_1+C_2}{2\pi}\). Substituting this into the CSA formula gives: \[ CSA = \pi \left( \frac{C_1 + C_2}{2\pi} \right) l = \frac{1}{2}(C_1 + C_2)l \] This formula allows us to calculate the CSA directly from the given circumferences and slant height.

Step 3: Detailed Explanation:
We are given: Slant height, \(l = 4\) cm.
Circumference of the larger end, \(C_1 = 18\) cm.
Circumference of the smaller end, \(C_2 = 6\) cm.
Using the direct formula for CSA: \[ CSA = \frac{1}{2}(C_1 + C_2)l \] Substitute the given values: \[ CSA = \frac{1}{2}(18 + 6) \times 4 \] \[ CSA = \frac{1}{2}(24) \times 4 \] \[ CSA = 12 \times 4 \] \[ CSA = 48 \text{ cm}^2 \]

Step 4: Final Answer:
The curved surface area of the frustum is 48 cm\(^2\).