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Question

A building is in the form of cylinder surmounted by a hemispherical dome. The base diameter of the dome is equal to 23 of the total height of the building. Find the height of the building if it contains 67121m3 of air.
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Solution

Height of building = height of cylinder + height of hemisphere
H=h+r
r is radius of cylinder and dome
diameter of dome =2/3× height of building
2r=23×(h+r)
6r=2h+2r 3r=h+r
h=2r
Volume of air in building = Volume of cylinder+ Volume of dome
67121=πr2h+23πr3
140821=πr2(h+23r)
=πr2(2r+2/3r)
=πr2(83r)
140821=83πr3
14087=8×227×r3
r3=140822×8=8
r=2 mh=4 m
Height of building =r+h=6 m

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