Question

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 880 / 21 cu. m of air. If the internal diameter of the building is equal to its total height above the floor, find the height of the building.

Answer 1

Let R be the radius, and h be the height of the cylindrical portion.

Height of the building, R+h = 2R = D = internal diameter.

⇒ h = R.

Volume = Volume of cylindrical portion + Volume of hemispherical portion

⇒ πR²h + (4πR³/3)/2 = 880/21

⇒(22/7)(R²h + 2R³/3) = 880/21

⇒R²(h+2R/3) = 440/3

⇒R²(R+2R/3) = 440/3

⇒R³(5/3) = 440/3

⇒R³ = 88

∴R = 4.45 m

and Total Height of the building, = R+h = 2R = 8.9 m

Answer 2

Let r be the radius, h be the height and d be the diameter.

Given that volume of the building = 880/21 cu.m.

Given that internal diameter of the building = total height above the floor.

  h = d = 2r ------ (1)

The volume of the air inside the building = volume of the cylindrical portion + volume of the hemisphere.

880/21 = pir^2h + (2/3)pir^3

880/21 = pir^3 + (2/3)pir^3 (h = r)

880/21 = 5/3 pir^3

r^3 = 880/21 * 7/22 * 3/5

r^3 = 18480/2310

r^3 = 8

r = 2.

Substitute r = 2 in (1), we get

h = 2 * r

   = 2 * 2

   = 4cm.


Therefore the height of the building = 4cm.


Hope this helps!