A bulinding is in the form of a cylinder surmounted by a hemispherical valuted 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 bulinding ?

Explanation

Let total height bulinding = Internal dimeter of the dome = 2 r cm

`therefore` Radius of bulinding (or dome ) `= (2r)/(2) = r m`
Height of cylinder `= 2r - r = rm`
`therefore`" " Volume of the cylinder `= pir^(2) (r) = pir^(3)m^(3)`
and volume of hemispherical dome cylinder `= (2)/(3) pir^(3) m^(3)`
`therefore` Total volume of the bulinding = Volume of the cylinder + Volume of hemispherical dome
`,, = (pi r^(3) +(2)/(3) pir^(3)) m^(3) = (5)/(3) pir^(3) m^(3)`
According to the question,
volume of the buliding = Volume of the air
`rArr" "(5)/(3) pir^(3) = 41(19)/(21)`
`rArr,, (5)/(3) pir^(3) = (880)/(21)`
`rArr" " r^(3) = (880 xx 7 xx 3)/(21xx22xx5) = (40 xx21)/(21xx5)= 8`
`rArr" " r^(3) = 8 rArr r= 2 `
` therefore` " " Height of the buliding `= 2r = 2 xx 2 = 4m`