# Mensuration-51

### From Homeworkwiki

**A circular hall, surmounted by a hemispherical roof, contains 5236 m ^{3} 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 r^{2} x r = Π r^{3} m^{3}

Volume of the hemispherical portion = 2/3 x Π r^{3} m^{3}

Therefore, Volume of the air in the hall = (Π r^{3} + 2/3 Π r^{3}) m^{3}

= 5/3 Π r^{3} m^{3}

Therefore, 5/3 Π r^{3} m^{3} = 5236

=> r^{3} = (5236 x 3) / (5 x 3.1416) = 1000 (approx)

=> r = 10

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