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A building is in the form of a cylinder surmounted by a hemisphere valted dome and contains 41\frac{19}{21} m^{3} of air.

If the internal diameter of dome is equal to its total height above the floor, find the height of the building.

Answer:

**Solution:**

As the building is the combination of cylinder and hemisphere so its volume will be the sum of volume of both the individual solids.

Let the radius of the dome be r.

Internal diameter = 2r

Given internal diameter is equal to total height.

Total height of the building = 2r

Height of the hemispherical area = r

So height of cylindrical area, h = 2r-r = r

Volume of the building = Volume of cylindrical area + volume of hemispherical area

=π r^{2}h + (2/3)π r^{3}

= π r^{3}+ (2/3)π r^{3} [∵h = r]

= π r^{3} (1+2/3)

=π r^{3} (3+2)/3

= (5/3)π r^{3}

**Given** Volume of the building = 41\frac{19}{21} =880/21

5/3)π r^{3}= 880/21

(5/3)×(22/7)×r^{3}= 880/21

r^{3} = 880×3×7/(5×22×21)

r^{3} = 880/110

r^{3} = 8

Taking cube root

r = 2 m

Height of the building = 2r = 2×2 = 4m

**Hence the height of the building is 4m.**

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