Question

A building is in the form of a cylinder surmounted by a hemispherical vaylted dome and contains 1144/21 m3 of air. If the internal diameter of the dome is equal to 4/5th of the total height above the floor, find the height of the building

Answer 1

v = 2/3 pi r^3 + pi r^2 h

d = 4/5 h, so r = 2h/5
but that's the total height. So, in terms of wall height, r = 2h/3

= pi r^2 (2r/3 + h)
1144/21 = pi (2h/3)^2 (2/3 * 2h/3 + h) = 52pi/81 h^3

h = 3 m

so, the wall is 3m high, and the dome is another 2m high, making the whole structure 5m high.