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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

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    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.

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