To find the amount of air (volume) present in the dome, you need to calculate the total surface area of the inner curved surface of the dome and then use that to determine the volume.
First, let's find the surface area of the dome's inner curved surface. You know that the cost of painting this surface is ₹49,280, and the rate is ₹40 per square meter. You can use this information to calculate the surface area.
Surface Area = Total Cost / Rate
Surface Area = ₹49,280 / ₹40/m^2
Surface Area = 1232 square meters
Now, let's find the volume of the dome. The inner curved surface of a dome can be thought of as a portion of a sphere. The formula for the volume of a sphere is (4/3)πr^3, where r is the radius. In your case, you need to find the radius of the dome.
The surface area of a hemisphere (half of a sphere) is 2πr^2. So, you have:
2πr^2 = 1232 square meters
Now, solve for r:
r^2 = 1232 square meters / (2π)
r^2 ≈ 196.35
r ≈ √196.35
r ≈ 14 meters (rounded to the nearest meter)
Now that you have the radius, you can calculate the volume of the dome:
Volume = (4/3)πr^3
Volume ≈ (4/3)π(14^3)
Volume ≈ (4/3)π(2744)
Volume ≈ 3658.67 cubic meters (rounded to two decimal places)
So, the amount of air present in the dome is approximately 3658.67 cubic meters.
Answers & Comments
Step-by-step explanation:
To find the amount of air (volume) present in the dome, you need to calculate the total surface area of the inner curved surface of the dome and then use that to determine the volume.
First, let's find the surface area of the dome's inner curved surface. You know that the cost of painting this surface is ₹49,280, and the rate is ₹40 per square meter. You can use this information to calculate the surface area.
Surface Area = Total Cost / Rate
Surface Area = ₹49,280 / ₹40/m^2
Surface Area = 1232 square meters
Now, let's find the volume of the dome. The inner curved surface of a dome can be thought of as a portion of a sphere. The formula for the volume of a sphere is (4/3)πr^3, where r is the radius. In your case, you need to find the radius of the dome.
The surface area of a hemisphere (half of a sphere) is 2πr^2. So, you have:
2πr^2 = 1232 square meters
Now, solve for r:
r^2 = 1232 square meters / (2π)
r^2 ≈ 196.35
r ≈ √196.35
r ≈ 14 meters (rounded to the nearest meter)
Now that you have the radius, you can calculate the volume of the dome:
Volume = (4/3)πr^3
Volume ≈ (4/3)π(14^3)
Volume ≈ (4/3)π(2744)
Volume ≈ 3658.67 cubic meters (rounded to two decimal places)
So, the amount of air present in the dome is approximately 3658.67 cubic meters.
Hope it's helpful
Mark as brainlest
[tex]Inner \: curved \: surface \: \\ area \: of \: dome = ₹ 49,280[/tex]
[tex]Rate \: of \: painting \: is \\ \\ = \: \: ₹ \: 40 \: per \: {m}^{2} [/tex]
[tex]Cost \: of \: painting = Rate \: \\ of \: painting × Surface \\ Area[/tex]
[tex]49,280 = 40 \times A \\ \\ A = \frac{49,280}{40} \\ \\ A = 1,232 \: sq \: m[/tex]
[tex]A= Represents \: the \: \\ surface \: area \: of \\ Hemisphere[/tex]
[tex]R= Radius \: of \: the \\ Hemisphere [/tex]
[tex]Surface \: area \: of \: \\ Hemi sphere = 2\pi {r}^{2} [/tex]
[tex]2\pi {r}^{2} = 1,232 \\ \\ {r}^{2} = \frac{1,232}{2\pi} \\ \\ {r}^{2} = 196.83 \\ \\ r = \sqrt{196.83} \\ \\ r = 14.03 \: m[/tex]
[tex]Volume \: of \: Hemisphere \\ \\ = \frac{2}{3} \pi {r}^{3} \\ \\ = \frac{2}{3} \times \frac{22}{7} \times ( {14.03)}^{3} \\ \\ = \frac{44}{21} \times 276.16 \\ \\ = \frac{12,151.04}{21} \\ \\ = 5,749.33 \: cm[/tex]