The internal and external diameters of a hollow hemispherical vessels are 24 cm and 25 cm repectively . The cost to paints ` 1 cm ^(2) ` of the surface of the vessels is Rs. 5 . Find the total cost to paint the vessel all over (use ` pi = 3 .14) `

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the given values
- Internal diameter of the vessel = 24 cm
- External diameter of the vessel = 25 cm
- Cost to paint 1 cm² = Rs. 5

### Step 2: Calculate the internal and external radii
- Internal radius (r) = Internal diameter / 2 = 24 cm / 2 = 12 cm
- External radius (R) = External diameter / 2 = 25 cm / 2 = 12.5 cm

### Step 3: Calculate the curved surface area of the hemispherical vessel
The formula for the curved surface area (CSA) of a hemisphere is:
\[ \text{CSA} = 2\pi r^2 \]

- Curved surface area of the outer hemisphere:
\[ \text{CSA}_{\text{outer}} = 2\pi R^2 = 2 \times 3.14 \times (12.5)^2 \]
\[ = 2 \times 3.14 \times 156.25 = 981.25 \, \text{cm}^2 \]

- Curved surface area of the inner hemisphere:
\[ \text{CSA}_{\text{inner}} = 2\pi r^2 = 2 \times 3.14 \times (12)^2 \]
\[ = 2 \times 3.14 \times 144 = 904.32 \, \text{cm}^2 \]

### Step 4: Calculate the area of the circular ring at the top
The area of the circular ring is given by:
\[ \text{Area}_{\text{ring}} = \pi (R^2 - r^2) \]
\[ = \pi (12.5^2 - 12^2) \]
\[ = 3.14 \times (156.25 - 144) \]
\[ = 3.14 \times 12.25 = 38.565 \, \text{cm}^2 \]

### Step 5: Calculate the total surface area to be painted
The total area to be painted is the sum of the outer curved surface area, the inner curved surface area, and the area of the ring:
\[ \text{Total Area} = \text{CSA}_{\text{outer}} + \text{CSA}_{\text{inner}} + \text{Area}_{\text{ring}} \]
\[ = 981.25 + 904.32 + 38.565 \]
\[ = 1924.135 \, \text{cm}^2 \]

### Step 6: Calculate the total cost to paint the vessel
The total cost is given by:
\[ \text{Total Cost} = \text{Total Area} \times \text{Cost per cm}^2 \]
\[ = 1924.135 \times 5 \]
\[ = 9620.675 \]

### Final Answer
The total cost to paint the vessel all over is approximately Rs. 9620.68.

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