Question

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains `41 19/21 m^3` of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building?

Answer 1

Let total height of the building = Internal diameter of the dome = 2r m

∴ Radius of building (or dome) = `(2r)/2` = r m

Height of cylinder = 2r – r = r m

∴ Volume of the cylinder = πr²(r) = πr³m³

And volume of hemispherical dome cylinder = `2/3 pir^3m^3`

∴ Total volume of the building

= Volume of the cylinder + Volume of hemispherical dome

= `(pir^3 + 2/3 pir^3)m^3`

= `5/3 pir^3m^3`

According to the question,

Volume of the building = Volume of the air

⇒ `5/3 pir^3 = 41 19/21`

⇒ `5/3 pir^3 = 880/21`

⇒ r³ = `(880 xx 7 xx 3)/(21 xx 22 xx 5)`

= `(40 xx 21)/(21 xx 5)`

= 8

⇒ r³ = 8

⇒ r = 2

∴ Height of the building = 2r

= 2 × 2

= 4 m