Question

The radii of the circular ends of a frustum of a cone are 14 cm and 8 cm. If the height of the frustum is 8 cm, find: (\(\pi = 3.14\))
(i) Curved surface area of frustum.
(ii) Total surface area of the frustum.
(iii) Volume of the frustum.

Hint:

Always start frustum problems by calculating the slant height \(l\), as it's needed for both curved and total surface area. Keep track of the two different radii, \(r_1\) and \(r_2\), and be careful with the order of operations in the volume formula.

Answer 1

Step 1: Understanding the Concept:
We need to apply the standard formulas for the curved surface area, total surface area, and volume of a frustum of a cone. The first step is to calculate the slant height.

Step 2: Key Formula or Approach:
Let \(r_1\) be the radius of the larger base, \(r_2\) be the radius of the smaller base, \(h\) be the height, and \(l\) be the slant height. \[\begin{array}{rl} \bullet & \text{Slant height: \(l = \sqrt{h^2 + (r_1 - r_2)^2}\)} \\ \bullet & \text{(i) Curved Surface Area (CSA): \(A_C = \pi (r_1 + r_2) l\)} \\ \bullet & \text{(ii) Total Surface Area (TSA): \(A_T = \pi (r_1 + r_2) l + \pi r_1^2 + \pi r_2^2\)} \\ \bullet & \text{(iii) Volume (V): \(V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2)\)} \\ \end{array}\]

Step 3: Detailed Explanation:
Given values: \[ r_1 = 14 \text{ cm, } r_2 = 8 \text{ cm, } h = 8 \text{ cm, } \pi = 3.14 \] First, calculate the slant height \(l\): \[ l = \sqrt{8^2 + (14 - 8)^2} = \sqrt{64 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ cm} \] (i) Curved surface area of frustum: \[ A_C = \pi (r_1 + r_2) l = 3.14 \times (14 + 8) \times 10 \] \[ A_C = 3.14 \times 22 \times 10 = 3.14 \times 220 = 690.8 \text{ cm}^2 \] (ii) Total surface area of the frustum: TSA = CSA + Area of top base + Area of bottom base \[ A_T = A_C + \pi r_1^2 + \pi r_2^2 \] \[ A_T = 690.8 + 3.14 \times (14^2) + 3.14 \times (8^2) \] \[ A_T = 690.8 + 3.14 \times 196 + 3.14 \times 64 \] \[ A_T = 690.8 + 615.44 + 200.96 \] \[ A_T = 1507.2 \text{ cm}^2 \] (iii) Volume of the frustum: \[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \] \[ V = \frac{1}{3} \times 3.14 \times 8 \times (14^2 + 8^2 + 14 \times 8) \] \[ V = \frac{25.12}{3} \times (196 + 64 + 112) \] \[ V = \frac{25.12}{3} \times (372) \] \[ V = 25.12 \times 124 \] \[ V = 3114.88 \text{ cm}^3 \]

Step 4: Final Answer:
(i) The curved surface area is 690.8 cm\(^2\).
(ii) The total surface area is 1507.2 cm\(^2\).
(iii) The volume is 3114.88 cm\(^3\).