A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs. 498.96. If the cost of white washing is Rs. 2.00 per square metre, find the (i) inside surface area of the dome. (ii) volume of the air inside the dome.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the total inside curved surface area of the dome.
The total cost of whitewashing is given as Rs. 498.96, and the cost of whitewashing is Rs. 2.00 per square meter.

To find the inside surface area (A) of the dome, we can use the formula:
\[ \text{Total Cost} = \text{Cost per square meter} \times \text{Surface Area} \]
Substituting the values:
\[ 498.96 = 2 \times A \]
Now, we can solve for A:
\[ A = \frac{498.96}{2} = 249.48 \, \text{m}^2 \]

### Step 2: Use the surface area to find the radius of the hemisphere.
The formula for the curved surface area (CSA) of a hemisphere is:
\[ \text{CSA} = 2\pi r^2 \]
Setting the CSA equal to the surface area we found:
\[ 2\pi r^2 = 249.48 \]
Now, we can solve for \( r^2 \):
\[ r^2 = \frac{249.48}{2\pi} \]
Using \( \pi \approx 3.14 \):
\[ r^2 = \frac{249.48}{2 \times 3.14} = \frac{249.48}{6.28} \approx 39.7 \]
Now, taking the square root to find \( r \):
\[ r \approx \sqrt{39.7} \approx 6.3 \, \text{m} \]

### Step 3: Calculate the volume of the air inside the dome.
The formula for the volume (V) of a hemisphere is:
\[ V = \frac{2}{3}\pi r^3 \]
Substituting the value of \( r \):
\[ V = \frac{2}{3} \pi (6.3)^3 \]
Calculating \( (6.3)^3 \):
\[ (6.3)^3 = 6.3 \times 6.3 \times 6.3 \approx 250.047 \]
Now substituting back into the volume formula:
\[ V = \frac{2}{3} \times 3.14 \times 250.047 \]
Calculating the volume:
\[ V \approx \frac{2}{3} \times 3.14 \times 250.047 \approx 523.91 \, \text{m}^3 \]

### Final Answers:
(i) The inside surface area of the dome is \( 249.48 \, \text{m}^2 \).

(ii) The volume of the air inside the dome is \( 523.91 \, \text{m}^3 \).

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