(i) Inside Surface Area of the Dome:
The cost of whitewashing is ₹20 per square meter. Given that the total cost of whitewashing is ₹4989.60, we can first calculate the total area whitewashed:
\[ \text{Area whitewashed} = \frac{\text{Total cost}}{\text{Cost per square meter}} = \frac{4989.60}{20} = 249.48 \text{ square meters} \]
The inside surface area of the hemisphere is the curved surface area (excluding the base). The formula for the surface area of a hemisphere is: \[ A = 2 \pi r^2 \] where \( r \) is the radius of the hemisphere. Since the total whitewashed area is equal to the inside surface area, we have: \[ 2 \pi r^2 = 249.48 \] Solving for \( r^2 \): \[ r^2 = \frac{249.48}{2 \pi} \approx \frac{249.48}{6.2832} \approx 39.7 \] Taking the square root to find \( r \): \[ r = \sqrt{39.7} \approx 6.31 \text{ meters} \] So, the radius of the dome is approximately \( 6.31 \) meters.
The inside surface area of the dome is \( 249.48 \, \text{square meters} \).
(ii) Volume of the Air Inside the Dome:
The volume \( V \) of a hemisphere is given by the formula: \[ V = \frac{2}{3} \pi r^3 \] Substituting \( r = 6.31 \) meters: \[ V = \frac{2}{3} \pi (6.31)^3 \] First, calculate \( (6.31)^3 \): \[ (6.31)^3 \approx 251.5 \] Now calculate the volume: \[ V = \frac{2}{3} \pi \times 251.5 \approx \frac{2}{3} \times 3.1416 \times 251.5 \approx \frac{2}{3} \times 789.25 \approx 526.17 \text{ cubic meters} \]
The volume of the air inside the dome is approximately \( 526.17 \, \text{cubic meters} \).