A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.

Answer

To find the capacity of the drinking glass in the shape of a frustum of a cone, we will use the formula for the volume of a frustum. The formula is given by:

\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]

where:
- \( V \) is the volume,
- \( h \) is the height of the frustum,
- \( r_1 \) is the radius of the larger circular end,
- \( r_2 \) is the radius of the smaller circular end.

### Step-by-Step Solution:

1. **Identify the dimensions:**
- The height \( h = 14 \) cm.
- The diameter of the larger end = 4 cm, so the radius \( r_1 = \frac{4}{2} = 2 \) cm.
- The diameter of the smaller end = 2 cm, so the radius \( r_2 = \frac{2}{2} = 1 \) cm.

2. **Substitute the values into the formula:**
\[ V = \frac{1}{3} \pi (14) (2^2 + 1^2 + 2 \cdot 1) \]

3. **Calculate \( r_1^2 \), \( r_2^2 \), and \( r_1 r_2 \):**
- \( r_1^2 = 2^2 = 4 \)
- \( r_2^2 = 1^2 = 1 \)
- \( r_1 r_2 = 2 \cdot 1 = 2 \)

4. **Combine these values:**
\[ r_1^2 + r_2^2 + r_1 r_2 = 4 + 1 + 2 = 7 \]

5. **Substitute back into the volume formula:**
\[ V = \frac{1}{3} \pi (14) (7) \]

6. **Calculate the volume:**
\[ V = \frac{1}{3} \pi (98) \]
\[ V = \frac{98}{3} \pi \text{ cm}^3 \]

7. **Using \( \pi \approx \frac{22}{7} \) for calculation:**
\[ V = \frac{98}{3} \cdot \frac{22}{7} \]
\[ V = \frac{98 \cdot 22}{3 \cdot 7} = \frac{2156}{21} \text{ cm}^3 \]

8. **Final calculation:**
\[ V \approx 102.67 \text{ cm}^3 \]

Thus, the capacity of the glass is approximately \( 102.67 \text{ cm}^3 \).