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A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains $41 \frac{19}{21}$ $m^{3}$ of air. If the internal diameter of the dome is equal to its total height above the floor, find the height of the building.

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8 m

5 m

4 m

None of these

answer is C.

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Detailed Solution

Let the total height of the building

=

internal diameter of the dome

= 2 r

Radius

= \frac{2 r}{2} = r

∴

Height of cylinder (h)

= 2 r - r = r

As we know that,
Volume of the hemispherical dome cylinder

= \frac{2}{3} {πr}^{3}

Volume of the cylinder =

π r^{2} h = {πr}^{3}

(h=r)

∴

Total volume of the building = Volume of the cylinder + Volume of hemispherical dome
=

\frac{2}{3} {πr}^{3} + {πr}^{3} = \frac{5}{3} π r^{3}

From the question we have,
Volume of the building = Volume of air

\Rightarrow \frac{5}{3} π r^{3} = 41 \frac{19}{21}

\Rightarrow \frac{5}{3} π r^{3} = \frac{880}{21}

\Rightarrow r^{3} = \frac{880 \times 7 \times 3}{21 \times 22 \times 5} = \frac{40 \times 21}{21 \times 5} = 8 m^{3} \Rightarrow r = 2 m

∴

Height of the building

= 2 r = 2 \times 2 = 4 m

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