Height of a drinking glass is 14 cm, the diameters of its two circular ends are 4 cm and 2 cm, then the capacity of glass is :

Question

Accepted Answer

To determine the capacity of the drinking glass, we need to calculate the volume of a frustum of a cone. The glass has two circular ends with diameters of 4 cm and 2 cm, and a height of 14 cm.

1. **Calculate the radii of the circular ends:**
   - The radius of the larger end (R) is half of the diameter: $ R = \frac{4}{2} = 2 \ \text{cm} $
   - The radius of the smaller end (r) is half of the diameter: $ r = \frac{2}{2} = 1 \ \text{cm} $

2. **Use the formula for the volume of a frustum of a cone:**
   $$
   V = \frac{1}{3} \pi h (R^2 + Rr + r^2)
   $$
   where $ h $ is the height of the frustum, $ R $ is the radius of the larger base, and $ r $ is the radius of the smaller base.

3. **Substitute the values into the formula:**
   $$
   V = \frac{1}{3} \pi (14) (2^2 + 2 \times 1 + 1^2)
   $$
   $$
   V = \frac{1}{3} \pi (14) (4 + 2 + 1)
   $$
   $$
   V = \frac{1}{3} \pi (14) \times 7
   $$
   $$
   V = \frac{1}{3} \times 22/7 \times 98
   $$
   $$
   V = \frac{1}{3} \times 308
   $$
   $$
   V = 102.67 \ \text{cm}^3
   $$

Thus, the capacity of the glass is $ 102.67 \ \text{cm}^3 $, which corresponds to Option 1.

**Explanation for why other options are incorrect:**

- **Option 2: $102.76 \ \text{cm}^3$** is incorrect because the calculated volume using the correct formula and values is $ 102.67 \ \text{cm}^3 $, not $ 102.76 \ \text{cm}^3 $.

- **Option 3: $202.66 \ \text{cm}^3$** is incorrect because it significantly overestimates the volume. The correct calculation yields $ 102.67 \ \text{cm}^3 $.

- **Option 4: $308.23 \ \text{cm}^3$** is incorrect as it is nearly three times the correct volume. The correct volume calculation results in $ 102.67 \ \text{cm}^3 $.

Therefore, Option 1 is the correct answer, as it accurately reflects the calculated capacity of the glass.