Question

There is a frustum of a cone whose height is 45 cm. If the radii of its ends are 28 cm and 7 cm, find its volume, curved surface area, and total surface area. (Use \( \pi = \frac{22}{7} \))

Hint:

When calculating the volume and surface area of a frustum, use the appropriate formulas and remember to find the slant height using the Pythagorean theorem.

Answer 1

Let the radii of the two ends of the frustum be \( r_1 = 28 \) cm and \( r_2 = 7 \) cm, and the height of the frustum be \( h = 45 \) cm.
The volume \( V \) of the frustum of a cone is given by the formula: \[ V = \frac{1}{3} \pi h \left( r_1^2 + r_1r_2 + r_2^2 \right). \] Substitute the given values into the formula: \[ V = \frac{1}{3} \times \frac{22}{7} \times 45 \left( 28^2 + 28 \times 7 + 7^2 \right). \] Calculate each term: \[ 28^2 = 784, \quad 28 \times 7 = 196, \quad 7^2 = 49. \] \[ V = \frac{1}{3} \times \frac{22}{7} \times 45 \left( 784 + 196 + 49 \right) = \frac{1}{3} \times \frac{22}{7} \times 45 \times 1029. \] Simplifying further: \[ V = \frac{1}{3} \times \frac{22}{7} \times 46305 = \frac{22 \times 46305}{21} = 48410 \, \text{cm}^3. \] Next, the curved surface area (CSA) of the frustum is given by: \[ \text{CSA} = \pi \left( r_1 + r_2 \right) l, \] where \( l \) is the slant height of the frustum. To find \( l \), we use the Pythagorean theorem: \[ l = \sqrt{(r_1 - r_2)^2 + h^2} = \sqrt{(28 - 7)^2 + 45^2} = \sqrt{21^2 + 45^2} = \sqrt{441 + 2025} = \sqrt{2466}. \] \[ l \approx 49.66 \, \text{cm}. \] Now, calculate the CSA: \[ \text{CSA} = \frac{22}{7} \times (28 + 7) \times 49.66 = \frac{22}{7} \times 35 \times 49.66 \approx 3854.6 \, \text{cm}^2. \] The total surface area (TSA) of the frustum is the sum of the curved surface area and the areas of the two circular ends: \[ \text{TSA} = \text{CSA} + \pi r_1^2 + \pi r_2^2. \] \[ \text{TSA} = 3854.6 + \frac{22}{7} \times (28^2 + 7^2) = 3854.6 + \frac{22}{7} \times (784 + 49) = 3854.6 + \frac{22}{7} \times 833. \] \[ \text{TSA} = 3854.6 + 2602 = 6456.6 \, \text{cm}^2. \]
Conclusion:
- The volume of the frustum is approximately \( 48410 \, \text{cm}^3 \).
- The curved surface area is approximately \( 3854.6 \, \text{cm}^2 \).
- The total surface area is approximately \( 6456.6 \, \text{cm}^2 \).