The internal and external diameters of a hollow hemi spherical vessel are 20 cm and 28 cm, respectively. Find the cost of painting the vessel at the rate of 14 paise per `cm^(2)`.

Answer

To solve the problem, we need to find the total cost of painting a hollow hemispherical vessel given its internal and external diameters. Here’s a step-by-step solution:

### Step 1: Determine the Radii
Given:
- Internal diameter = 20 cm
- External diameter = 28 cm

To find the radii:
- Internal radius (r) = Internal diameter / 2 = 20 cm / 2 = 10 cm
- External radius (R) = External diameter / 2 = 28 cm / 2 = 14 cm

### Step 2: Calculate the Curved Surface Area (CSA) of the Hemispheres
The formula for the curved surface area of a hemisphere is:
\[ \text{CSA} = 2\pi r^2 \]

For the external hemisphere:
\[ \text{CSA}_{\text{outer}} = 2\pi R^2 = 2\pi (14^2) = 2\pi (196) = 392\pi \]

For the internal hemisphere:
\[ \text{CSA}_{\text{inner}} = 2\pi r^2 = 2\pi (10^2) = 2\pi (100) = 200\pi \]

### Step 3: Calculate the Area of the Circular Base
The area of the circular base that needs to be painted is the difference between the areas of the outer and inner circles:
\[ \text{Area}_{\text{base}} = \pi R^2 - \pi r^2 = \pi (14^2) - \pi (10^2) = \pi (196 - 100) = 96\pi \]

### Step 4: Calculate the Total Area to be Painted
The total area to be painted is the sum of the curved surface areas and the area of the base:
\[ \text{Total Area} = \text{CSA}_{\text{outer}} + \text{CSA}_{\text{inner}} + \text{Area}_{\text{base}} \]
\[ \text{Total Area} = 392\pi + 200\pi + 96\pi = 688\pi \]

### Step 5: Substitute the Value of π
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Total Area} = 688 \times \frac{22}{7} \]
Calculating this gives:
\[ \text{Total Area} \approx 688 \times 3.14 \approx 2162.29 \, \text{cm}^2 \]

### Step 6: Calculate the Cost of Painting
Given the cost of painting is 14 paise per cm²:
\[ \text{Cost} = \text{Total Area} \times \text{Cost per cm}^2 \]
\[ \text{Cost} = 2162.29 \times 14 \, \text{paise} \]
\[ \text{Cost} = 30272.06 \, \text{paise} \]

### Step 7: Convert Paise to Rupees
Since 1 rupee = 100 paise:
\[ \text{Cost in Rupees} = \frac{30272.06}{100} = 302.72 \, \text{rupees} \]

### Final Answer
The cost of painting the vessel is approximately **302.72 rupees**.

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