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A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41 19/21 m³ of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building

Solution:

Given, a building is in the form of a cylinder surmounted by a hemispherical vaulted dome.

Building contains 41 19/21 m³ of air.

Internal diameter of hemispherical dome = total height of the building

We have to find the height of the building.

Volume of building = volume of cylinder + volume of hemisphere

Let the total height of the building be h

Given, Diameter = h

Radius = h/2

Height of cylinder = h - h/2 = h/2

Volume of cylinder = πr²h

Volume = π(h/2)²(h/2)

= πh³/8 m³

Volume of hemisphere = (2/3)πr³

= (2/3)π(h/2)³

= 2πh³/24 m³

Given, volume of air in the building = 41 19/21 m³

= [41(21) + 19]/21

= [861 + 19]/21

= 880/21 m³

Now, 880/21 = πh³/8 + 2πh³/24

880/21 = (3πh³ + 2πh³)/24

880/21 = 5πh³/24

Solving for h,

h³ = 880(24)/21(5π)

h³ = 64

Taking cube root,

h = 4 m

Therefore, the height of the building is 4 m.

✦ Try This: A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 42 17/31 m³ of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building?

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 13

NCERT Exemplar Class 10 Maths Exercise 12.4 Problem 15

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41 19/21 m³ of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building

Summary:

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41 19/21 m³ of air. If the internal diameter of dome is equal to its total height above the floor, the height of the building is 4 m

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