The internal and external diameters of a hollow hemispherical vessel are 42 cm and 45 cm respectively. Find its capacity and also its outer curved surface area.

Answer

To solve the problem of finding the capacity and the outer curved surface area of a hollow hemispherical vessel with given internal and external diameters, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Values:**
- Internal diameter (d) = 42 cm
- External diameter (D) = 45 cm

2. **Calculate the Internal and External Radii:**
- Internal radius (r) = d/2 = 42/2 = 21 cm
- External radius (R) = D/2 = 45/2 = 22.5 cm

3. **Calculate the Volume of the Hollow Hemisphere:**
- The formula for the volume of a hollow hemisphere is:
\[ V = \frac{2}{3} \pi (R^3 - r^3) \]
- Substitute the values of R and r into the formula:
\[ V = \frac{2}{3} \pi (22.5^3 - 21^3) \]

4. **Calculate \( R^3 \) and \( r^3 \):**
- \( R^3 = 22.5^3 = 11390.625 \)
- \( r^3 = 21^3 = 9261 \)

5. **Subtract \( r^3 \) from \( R^3 \):**
- \( R^3 - r^3 = 11390.625 - 9261 = 2129.625 \)

6. **Substitute Back to Find Volume:**
- Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{2}{3} \times \frac{22}{7} \times 2129.625 \]
- Calculate:
\[ V = \frac{44}{21} \times 2129.625 \]
- \( V \approx 4462.07 \, \text{cm}^3 \)

7. **Calculate the Outer Curved Surface Area:**
- The formula for the curved surface area of a hemisphere is:
\[ A = 2 \pi R^2 \]
- Substitute the value of R:
\[ A = 2 \pi (22.5^2) \]

8. **Calculate \( R^2 \):**
- \( R^2 = 22.5^2 = 506.25 \)

9. **Substitute Back to Find Area:**
- Using \( \pi \approx \frac{22}{7} \):
\[ A = 2 \times \frac{22}{7} \times 506.25 \]
- Calculate:
\[ A = \frac{44}{7} \times 506.25 \approx 3182.14 \, \text{cm}^2 \]

### Final Answers:
- Capacity of the vessel: \( 4462.07 \, \text{cm}^3 \)
- Outer curved surface area: \( 3182.14 \, \text{cm}^2 \)