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# A building is in the form of a cylinder surmounted by a hemispherical dome (see Fig. 12.12). The base diameter of the dome is equal to 2/3 of the total height of the building. Find the height of the building, if it contains 67 1/21 m³ of air

**Solution:**

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

The base diameter of the dome = (2/3)total height of the building.

We have to find the height of the building, if it contains 67 1/21 m³ of air.

From the figure,

Let r be the radius of the dome

Let h be the height of the building

Let H be the height of the cylinder.

Given, 2r = (2/3)h

r = h/3 m

From the figure,

H = h - r

H = h - h/3

H = 2h/3 m

Volume of air inside the building = volume of air inside the dome + volume of air inside the cylinder.

Volume of air in dome = (2/3)πr³

= (2/3)π(h/3)³

= 2h³π/81 m³

Volume of air in cylinder = πr²H

= π(h/3)²(2h/3)

= 2πh³/27 m³

Volume of air inside the building = 2h³π/81 + 2πh³/27

= (2h³π + 3(2h³π))/81

= 8h³π/81

Given, volume of air inside the building = 67 1/21 m³

[67(21) + 1]/21 = 8h³π/81

1408/21 = 8h³π/81

Solving for h,

h³ = 1408(81) / 21(8π)

h³ = 216

Taking cube root,

h = 6 m

Therefore, the height of the building is 6 m.

**✦ Try This: **A building is in the form of a cylinder surmounted by a hemispherical dome. The base diameter of the dome is equal to 1/3 of the total height of the building. Find the height of the building, if it contains 76 1/19 m³ of air.

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

**NCERT Exemplar Class 10 Maths Exercise 12.4 Sample Problem 3**

## A building is in the form of a cylinder surmounted by a hemispherical dome (see Fig. 12.12). The base diameter of the dome is equal to 2/3 of the total height of the building. Find the height of the building, if it contains 67 1/21 m³ of air

**Summary: **

A building is in the form of a cylinder surmounted by a hemispherical dome (see Fig. 12.12). The base diameter of the dome is equal to 2/3 of the total height of the building. The height of the building, if it contains 67 1/21 m³ of air, is 6 m

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