The internal and external diameters of a hollow hemispherical vessel are 21 cm and 25.2 cm respectively. Find the cost of painting it, all over, at the rate of 1.50 per `cm^(2).`

Answer

To solve the problem of finding the cost of painting a hollow hemispherical vessel with given internal and external diameters, we will follow these steps:

### Step 1: Determine the Internal and External Radii
- The internal diameter is given as 21 cm, so the internal radius \( r_1 \) is:
\[ r_1 = \frac{21}{2} = 10.5 \text{ cm} \]
- The external diameter is given as 25.2 cm, so the external radius \( r_2 \) is:
\[ r_2 = \frac{25.2}{2} = 12.6 \text{ cm} \]

### Step 2: Calculate the Internal Curved Surface Area
- The formula for the curved surface area of a hemisphere is:
\[ \text{Curved Surface Area} = 2\pi r^2 \]
- For the internal surface area, we substitute \( r_1 \):
\[ \text{Internal Curved Surface Area} = 2 \times \frac{22}{7} \times (10.5)^2 \]
- Calculating \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]
- Now substituting back:
\[ \text{Internal Curved Surface Area} = 2 \times \frac{22}{7} \times 110.25 \]
\[ = \frac{44}{7} \times 110.25 \]
- Calculating this gives:
\[ = \frac{44 \times 110.25}{7} = \frac{4851}{7} \approx 693 \text{ cm}^2 \]

### Step 3: Calculate the External Curved Surface Area
- Now we calculate the external surface area using \( r_2 \):
\[ \text{External Curved Surface Area} = 2 \times \frac{22}{7} \times (12.6)^2 \]
- Calculating \( (12.6)^2 \):
\[ (12.6)^2 = 158.76 \]
- Now substituting back:
\[ \text{External Curved Surface Area} = 2 \times \frac{22}{7} \times 158.76 \]
\[ = \frac{44}{7} \times 158.76 \]
- Calculating this gives:
\[ = \frac{6984.48}{7} \approx 997.92 \text{ cm}^2 \]

### Step 4: Calculate Total Surface Area to be Painted
- The total surface area to be painted is the sum of the internal and external curved surface areas:
\[ \text{Total Surface Area} = \text{Internal Curved Surface Area} + \text{External Curved Surface Area} \]
\[ = 693 + 997.92 = 1690.92 \text{ cm}^2 \]

### Step 5: Calculate the Cost of Painting
- The cost of painting per cm² is given as 1.50. Therefore, the total cost is:
\[ \text{Cost} = \text{Total Surface Area} \times \text{Cost per cm}^2 \]
\[ = 1690.92 \times 1.50 = 2536.38 \text{ rupees} \]

### Final Answer
The cost of painting the hollow hemispherical vessel is **2536.38 rupees**.

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