Answer: 5749.33 cm

Step-by-step explanation:

To find the amount of air present in the dome, we first need to calculate the total surface area of the inner curved surface of the dome. We're given that the cost to paint this surface is ₹49,280 at a rate of ₹40 per square meter.

Step 1: Find the Surface Area of the Dome

Let's assume the radius of the dome is r meters. The inner curved surface area of a dome is given by the formula: [tex]\[ \text{Surface Area} = 2 \times \pi \times r^2 \][/tex]

Step 2: Find the Radius of the Dome

Now we can find the radius of the dome using the formula for the surface area: 1232=2×π×r^2

[tex]Solving for \( r^2 \):\[ r^2 = \frac{1232}{2\pi} \]\[ r^2 \approx 196.51 \]\[ r \approx \sqrt{196.51} \]\[ r \approx 14 \, \text{meters} \][/tex]

To find the amount of air present in the dome, we first need to calculate the total surface area of the inner curved surface of the dome. We're given that the cost to paint this surface is ₹49,280 at a rate of ₹40 per square meter.

**Step 1: Find the Surface Area of the Dome**

Let's assume the radius of the dome is r meters. The inner curved surface area of a dome is given by the formula:

[tex]\[ \text{Surface Area} = 2 \times \pi \times r^2 \][/tex]

We are given that the cost to paint this surface is ₹49,280 at a rate of ₹40 per square meter. So, the surface area in square meters is:

[tex]\[ \text{Surface Area} = \frac{\text{Cost}}{\text{Rate}} = \frac{49280}{40} = 1232 \, \text{m}^2 \][/tex]

**Step 2: Find the Radius of the Dome**

Now we can find the radius of the dome using the formula for the surface area:

[tex]\[ 1232 = 2 \times \pi \times r^2 \]Solving for \( r^2 \):\[ r^2 = \frac{1232}{2\pi} \]\[ r^2 \approx 196.51 \]\[ r \approx \sqrt{196.51} \]\[ r \approx 14 \, \text{meters} \][/tex]

**Step 3: Find the Volume of the Dome**

The volume of a dome is given by the formula:

[tex]\[ \text{Volume} = \frac{2}{3} \times \pi \times r^3 \]\[ \text{Volume} = \frac{2}{3} \times \pi \times 14^3 \]\[ \text{Volume} \approx 5749.33 \, \text{cubic meters} \]To convert this volume from cubic meters to cubic centimeters (1 cubic meter = \(1 \times 10^6\) cubic centimeters):\[ \text{Volume} \approx 5749.33 \times 10^6 \, \text{cubic centimeters} \]So, the amount of air present in the dome is approximately \(5749.33 \, \text{cm}^3\), which matches the correct answer given.[/tex]

[tex]Curved \: surface \: area \: of \\ cylinder = 2\pi rh[/tex]

[tex]49,280 = 2 \times \frac{22}{7} \times r \times 40 \\ \\ 49,280 = \frac{44r \times 40}{7} \\ \\ (49,280 \times 7) = 1,760 \: r \\ \\ 3,44,960 = 1,760 \: r \\ \\ r = \frac{3,44,960}{1,760} \\ \\ r = 196 \: {cm}^{2} [/tex]