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. None of these

Answer

To solve the problem step by step, we need to find the surface area of the hemispherical dome and then calculate the cost of painting it based on the given rate.

### 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. We can rearrange this formula to find \( r \):
\[ r = \frac{C}{2\pi} \]
Substituting the value of \( C \):
\[ r = \frac{17.6}{2\pi} \]
Using \( \pi \approx 3.14 \):
\[ r = \frac{17.6}{2 \times 3.14} \approx \frac{17.6}{6.28} \approx 2.8 \text{ m} \]

### Step 2: Calculate the curved surface area of the hemispherical dome
The formula for the curved surface area \( A \) of a hemisphere is:
\[ A = 2\pi r^2 \]
Substituting the value of \( r \):
\[ A = 2\pi (2.8)^2 \]
Calculating \( (2.8)^2 \):
\[ (2.8)^2 = 7.84 \]
Now substituting back:
\[ A = 2\pi \times 7.84 \approx 2 \times 3.14 \times 7.84 \approx 49.28 \text{ m}^2 \]

### Step 3: Convert the area from square meters to square centimeters
Since the cost of painting is given per 100 cm², we need to convert the area from m² to cm².
1 m² = 10,000 cm², so:
\[ 49.28 \text{ m}^2 = 49.28 \times 10,000 \text{ cm}^2 = 492800 \text{ cm}^2 \]

### Step 4: Calculate the cost of painting
The cost of painting is given as Rs. 8 per 100 cm². First, we find the number of 100 cm² units in 492800 cm²:
\[ \text{Number of 100 cm}^2 = \frac{492800}{100} = 4928 \]
Now, calculating the total cost:
\[ \text{Total Cost} = 4928 \times 8 = 39424 \text{ Rs.} \]

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

---