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A building is in the form of a cylinder surmounted by a hemispherical dome (see the figure). The base radius of the dome is equal to $13$ of the total height of the building. What is the height of the building, if it contains $67121 m 3$ of air?

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answer is A.

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Detailed Solution

Given that the base radius of the dome is

13

of the total height of the building and the building contains

67121 m 3

of air.

We have the following formulae,
Volume of cylinder

= π r 2 h

Volume of hemisphere

= 23 π r 3

Radius of dome will be,

= 13 × H

Total height of the building

r = H 3 h = H − r = H − H 3 = 2 H 3

The volume of the air inside the building = volume of air inside the done + volume of air inside the cylinder

= 23 × π × H 3 3 + π × H 3 2 × 23 H = 23 π × H 3 27 + π × H 2 9 × 2 H 3 = 8 π H 3 81 m 2

The height of the building will be,

67121 m 3 = 8 π H 3 81 67121 = 8 π H 3 81 H 3 = 67 × 21 + 1 21 × 81 × 7 8 × 22 H 3 = 216 H = 6 m

Thus, the height of the building is 6m.
Therefore the correct option is 1.

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