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.
(i) Curved surface area of frustum.
(ii) Total surface area of the frustum.
(iii) Volume of the frustum.
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Solution and Explanation
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\(^3\).
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