A spherical glass vessel has a cylindrical neck 7 cm long and 4 cm in diameter. The diameter of the spherical part is 21 cm. Find the quantity

Answer

To find the total volume of the spherical glass vessel with a cylindrical neck, we need to calculate the volume of both the cylinder and the sphere separately and then add them together.

### Step-by-Step Solution:

1. **Identify the dimensions:**
- The cylindrical neck has a height (h) of 7 cm and a diameter of 4 cm.
- The spherical part has a diameter of 21 cm.

2. **Calculate the radius of the cylinder:**
- Radius (r1) of the cylindrical neck = Diameter / 2 = 4 cm / 2 = 2 cm.

3. **Calculate the volume of the cylinder:**
- The formula for the volume of a cylinder is:
\[ V_{cylinder} = \pi r_1^2 h \]
- Substituting the values:
\[ V_{cylinder} = \pi (2)^2 (7) = \pi (4)(7) = 28\pi \text{ cm}^3 \]

4. **Calculate the radius of the sphere:**
- Radius (r2) of the spherical part = Diameter / 2 = 21 cm / 2 = 10.5 cm.

5. **Calculate the volume of the sphere:**
- The formula for the volume of a sphere is:
\[ V_{sphere} = \frac{4}{3} \pi r_2^3 \]
- Substituting the radius:
\[ V_{sphere} = \frac{4}{3} \pi (10.5)^3 \]
- First, calculate \( (10.5)^3 \):
\[ (10.5)^3 = 10.5 \times 10.5 \times 10.5 = 1157.625 \]
- Now substituting this value:
\[ V_{sphere} = \frac{4}{3} \pi (1157.625) \approx \frac{4}{3} \times \frac{22}{7} \times 1157.625 \]

6. **Calculate the total volume:**
- Now, add the volumes of the cylinder and the sphere:
\[ V_{total} = V_{cylinder} + V_{sphere} \]
- Substitute the calculated values:
\[ V_{total} = 28\pi + \frac{4}{3} \pi (1157.625) \]
- To simplify, we can calculate \( \frac{4}{3} \times \frac{22}{7} \times 1157.625 \) and then add it to \( 28\pi \).

7. **Final Calculation:**
- Calculate \( V_{sphere} \):
\[ V_{sphere} \approx 4 \times 22 \times 1157.625 / (3 \times 7) \approx 4 \times 22 \times 165.375 \approx 14552.25 \text{ cm}^3 \]
- Therefore, the total volume:
\[ V_{total} \approx 28\pi + 14552.25 \approx 4939 \text{ cm}^3 \]

### Final Answer:
The total volume of the spherical glass vessel is approximately **4939 cm³**.