Question

A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains `1408/21 m^3` of air, find the height of the cylindrical part. `("Use"  pi = 22/7)`

Answer 1

Volume of the cylinder: V_cylinder = πr²h

Volume of the hemisphere: V_hemisphere = `2/3pir^3`

`pir^2h + 2/3 pir^3 = 1408/21`

`pi(h/2)^2 h + 2/3 pi (h/2)^3 = 1408/21`

`pi h^2/4 h + 2/3pi h^3/8 = 1408/21`

`(pih^3)/4 + (2pih^3)/24 = 1408/21`

`(pih^3)/4 + (pih^3)/12 = 1408/21`

`(3pih^3)/12 + (pih^3)/12 = 1408/12`

`(4pih^3)/12 = 1408/21`

`(pih^3)/3 = 1408/21`

`(22/7 h^3)/3 = 1408/21`

`(22h^3)/21 = 1408/21`

22h³ = 1408

`h^3 = 1408/22`

h³ = 64

`h = root3 64`

h = 4 m