A hemispherical dome of a building needs to be painted from outside. If the circumference of the base of the dome is 17.6 m, then find the cost of painting it at the rate of Rs.8 per 100 `cm^2`

Options

A. Rs. 35680
B. Rs. 28650
C. Rs. 39424
D. Rs. 40524

Answer

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

### Step 1: Find the radius of the hemispherical dome.
The circumference (C) of the base of the dome is given as 17.6 m. The formula for the circumference of a circle is:
\[ C = 2\pi r \]
where \( r \) is the radius.

Rearranging the formula to find the radius:
\[ r = \frac{C}{2\pi} \]

Substituting the value of C:
\[ r = \frac{17.6}{2\pi} \]

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

Now, substitute the value of \( r \) from Step 1 into this formula to find the surface area.

### Step 3: Convert the surface area into square centimeters.
Since the cost of painting is given per 100 cm², we need to convert the surface area from m² to cm².
1 m² = 10,000 cm², so multiply the surface area in m² by 10,000 to get it in cm².

### Step 4: Calculate the cost of painting.
The cost of painting is given as Rs. 8 per 100 cm². To find the total cost, we will use the formula:
\[ \text{Total Cost} = \left( \frac{\text{Surface Area in cm²}}{100} \right) \times 8 \]

### Step 5: Final Calculation.
Perform the calculations to find the total cost of painting the hemispherical dome.

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### Step-by-Step Solution:

1. **Calculate the radius:**
\[ r = \frac{17.6}{2\pi} \approx \frac{17.6}{6.28} \approx 2.8 \text{ m} \]

2. **Calculate the curved surface area:**
\[ \text{CSA} = 2\pi r^2 = 2\pi (2.8)^2 = 2\pi (7.84) \approx 49.28 \text{ m}^2 \]

3. **Convert the surface area to cm²:**
\[ \text{CSA in cm²} = 49.28 \times 10,000 = 492800 \text{ cm}^2 \]

4. **Calculate the total cost of painting:**
\[ \text{Total Cost} = \left( \frac{492800}{100} \right) \times 8 = 4928 \times 8 = 39424 \text{ Rs} \]

### Final Answer:
The cost of painting the hemispherical dome is Rs. 39,424.

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