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A circular hall, surmounted by a hemispherical roof, contains 5236 m3 of air and the internal diameter
of the room is equal to the height of the highest point of the roof from the floor. Find the height.

Solution: Let the radius of the floor be r m

Therefore, Volume of the cylindrical portion = Π r2 x r = Π r3 m3

Volume of the hemispherical portion = 2/3 x Π r3 m3

Therefore, Volume of the air in the hall = (Π r3 + 2/3 Π r3) m3

= 5/3 Π r3 m3

Therefore, 5/3 Π r3 m3 = 5236

=> r3 = (5236 x 3) / (5 x 3.1416) = 1000 (approx)

=> r = 10

Therefore, Height of the room = 2 r = 20 m approx

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