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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