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

Options

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

Explanation

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.

### Step 1: Find the radius of the hemisphere.
The circumference of the base of the dome is given as 17.6 m. The formula for the circumference (C) of a circle is:

\[ C = 2\pi r \]

where \( r \) is the radius. We can rearrange this formula to find the radius:

\[ r = \frac{C}{2\pi} \]

Substituting the given circumference:

\[ r = \frac{17.6}{2\pi} \]

Calculating this gives:

\[ r \approx \frac{17.6}{6.28} \approx 2.8 \text{ m} \]

### Step 2: Calculate the surface area of the hemispherical dome.
The surface area (A) of a hemisphere is given by the formula:

\[ A = 2\pi r^2 \]

Substituting the radius we found:

\[ A = 2\pi (2.8)^2 \]

Calculating \( (2.8)^2 \):

\[ (2.8)^2 = 7.84 \]

Now substituting back into the area formula:

\[ A = 2\pi \times 7.84 \approx 2 \times 3.14 \times 7.84 \approx 49.2 \text{ m}^2 \]

### Step 3: Convert the area to cm².
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:

\[ A = 49.2 \text{ m}^2 \times 10,000 \text{ cm}^2/\text{m}^2 = 492,000 \text{ cm}^2 \]

### Step 4: Calculate the cost of painting.
The cost of painting is Rs 8 per 100 cm². First, we need to find out how many 100 cm² are in 492,000 cm²:

\[ \text{Number of 100 cm}^2 = \frac{492,000}{100} = 4920 \]

Now, we can calculate the total cost:

\[ \text{Total Cost} = 4920 \times 8 = 39,360 \text{ Rs} \]

### Final Answer:
The cost of painting the hemispherical dome is Rs 39,360.

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