UPSC  >  NCERT Solutions: Surface Areas & Volumes (Exercise 11.4)

# Surface Areas & Volumes (Exercise 11.4) NCERT Solutions - NCERT Textbooks (Class 6 to Class 12) - UPSC

Q1. Find the volume of a sphere whose radius is
(i) 7 cm
(ii) 0.63 m (Assume π = 22 / 7)
Ans: (i)
Radius of sphere, r = 7 cm
Using, Volume of sphere = (4 / 3) πr3
= (4 / 3) × (22 / 7) × 73 = 4312 / 3
Hence, volume of the sphere is 4312 / 3 cm3
(ii) Radius of sphere, r = 0.63 m
Using, volume of sphere = (4 / 3) πr3
= (4 / 3) × (22 / 7) × 0.633 = 1.0478
Hence, volume of the sphere is 1.05 m3 (approx).

Q2. Find the amount of water displaced by a solid spherical ball of diameter
(i) 28 cm
(ii) 0.21 m (Assume π = 22 / 7)
Ans: (i)
Diameter = 28 cm
Radius, r = 28 / 2 cm = 14cm
Volume of the solid spherical ball = (4 / 3) πr3
Volume of the ball = (4 / 3) × (22 / 7) × 143 = 34496 / 3
Hence, volume of the ball is 34496 / 3 cm3
(ii) Diameter = 0.21 m
Radius of the ball = 0.21 / 2 m = 0.105 m
Volume of the ball = (4 / 3)πr3
Volume of the ball = (4 / 3) × (22 / 7) × 0.1053
Hence, volume of the ball = 0.004851 m3

Q3. The diameter of a metallic ball is 4.2cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm3? (Assume π = 22 / 7)
Ans:
Given,
Diameter of a metallic ball = 4.2 cm
Radius(r) of the metallic ball, r = 4.2 / 2 cm = 2.1 cm
Volume formula = 4 / 3 πr3
Volume of the metallic ball = (4 / 3) × (22 / 7) × 2.13
Volume of the metallic ball = 38.808 cm3
Now, using relationship between, density, mass and volume,
Density = Mass / Volume
Mass = Density × volume
= (8.9 × 38.808) g
= 345.3912 g
Mass of the ball is 345.39 g (approx).

Q4. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
Ans:
Let the diameter of earth be “d”. Therefore, the radius of earth will be will be d/2
Diameter of moon will be d / 4 and the radius of moon will be d/8
Find the volume of the moon:
Volume of the moon = (4 / 3) πr3 = (4 / 3) π (d / 8)3 = 4 / 3π(d3 / 512)
Find the volume of the earth:
Volume of the earth = (4 / 3) πr3 = (4 / 3) π (d / 2)3 = 4 / 3π(d3 / 8)
Fraction of the volume of the earth is the volume of the moon

Volume of moon is of the 1 / 64 volume of earth.

Q5. How many litres of milk can a hemispherical bowl of diameter 10.5cm hold? (Assume π = 22 / 7)
Ans:
Diameter of hemispherical bowl = 10.5 cm
Radius of hemispherical bowl, r = 10.5 / 2 cm = 5.25 cm
Formula for volume of the hemispherical bowl = (2 / 3) πr3
Volume of the hemispherical bowl = (2 / 3) × (22 / 7) × 5.253 = 303.1875
Volume of the hemispherical bowl is 303.1875 cm3
Capacity of the bowl = (303.1875) / 1000 L = 0.303 litres(approx.)
Therefore, hemispherical bowl can hold 0.303 litres of milk.

Q6. A hemi spherical tank is made up of an iron sheet 1cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank. (Assume π = 22/7)
Ans:
Inner Radius of the tank, (r) = 1m
Outer Radius (R) = 1.01m
Volume of the iron used in the tank = (2 / 3) π (R3 – r3)
Put values,
Volume of the iron used in the hemispherical tank = (2 / 3) × (22 / 7) × (1.013 – 13) = 0.06348
So, volume of the iron used in the hemispherical tank is 0.06348 m3.

Q7. Find the volume of a sphere whose surface area is 154 cm2. (Assume π = 22 / 7)
Ans:
Let r be the radius of a sphere.
Surface area of sphere = 4πr2
4πr2 = 154 cm2 (given)
r2 = (154 × 7) / (4 × 22) r = 7 / 2
Radius is 7 / 2 cm
Now, Volume of the sphere = (4 / 3) πr3
Volume of the sphere = (4 / 3) x (22 / 7) x (7 / 2)3 = 179 x 2 / 3
Volume of the sphere is 179 x 2 / 3 cm3

Q8. A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs. 4989.60. If the cost of white-washing isRs20 per square meter, find the
(i) Inside surface area of the dome
(ii) volume of the air inside the dome (Assume π = 22 / 7)
Ans: (i)
Cost of white-washing the dome from inside = Rs 4989.60
Cost of white-washing 1marea = Rs 20
CSA of the inner side of dome = 498.96 / 2 m2  = 249.48 m2
(ii) Let the inner radius of the hemispherical dome be r.
CSA of inner side of dome = 249.48 m2 (from (i))
Formula to find CSA of a hemi sphere = 2πr2
2πr2 = 249.48
2 × (22 / 7) × r2 = 249.48
r2 = (249.48 × 7) / (2 × 22)
r2 = 39.69
r = 6.3
So, radius is 6.3 m
Volume of air inside the dome = Volume of hemispherical dome
Using formula, volume of the hemisphere
= 2 / 3 πr3 = (2 / 3) × (22 / 7) × 6.3 × 6.3 × 6.3
= 523.908
= 523.9(approx.)
Volume of air inside the dome is 523.9 m3.

Q9. Twenty-seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S’. Find the
(i) radius r’ of the new sphere,
(ii) ratio of Sand S’.
Ans:
Volume of the solid sphere = (4 / 3)πr3
Volume of twenty seven solid sphere = 27 × (4 / 3)πr3 = 36 πr3
(i) New solid iron sphere radius = r’
Volume of this new sphere = (4/3)π(r’)3
(4 / 3)π(r’)3 = 36 πr3
(r’)3 = 27r3
r’= 3r
Radius of new sphere will be 3r (thrice the radius of original sphere)
(ii) Surface area of iron sphere of radius r, S = 4πr2
Surface area of iron sphere of radius r’= 4π (r’)2
Now
S / S’ = (4πr2) / ( 4π (r’)2)
S / S’ = r2 / (3r’)2 = 1 / 9
The ratio of S and S’ is 1: 9.

Q10. A capsule of medicine is in the shape of a sphere of diameter 3.5mm. How much medicine (in mm3) is needed to fill this capsule? (Assume π = 22 / 7)
Ans:
Diameter of capsule = 3.5 mm
Radius of capsule, say r = diameter / 2 = (3.5 / 2) mm = 1.75mm
Volume of spherical capsule = 4 / 3 πr3
Volume of spherical capsule = (4 / 3) × (22 / 7) × (1.75)3 = 22.458
The volume of the spherical capsule is 22.46 mm3.

The document Surface Areas & Volumes (Exercise 11.4) NCERT Solutions | NCERT Textbooks (Class 6 to Class 12) - UPSC is a part of the UPSC Course NCERT Textbooks (Class 6 to Class 12).
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## FAQs on Surface Areas & Volumes (Exercise 11.4) NCERT Solutions - NCERT Textbooks (Class 6 to Class 12) - UPSC

 1. What is the formula for finding the surface area of a cylinder?
Ans. The formula for finding the surface area of a cylinder is given by 2πr(r + h), where r is the radius of the base and h is the height of the cylinder.
 2. How do you calculate the volume of a sphere?
Ans. The volume of a sphere can be calculated using the formula (4/3)πr^3, where r is the radius of the sphere.
 3. How can we find the surface area of a cone?
Ans. The surface area of a cone can be found using the formula πr(r + l), where r is the radius of the base and l is the slant height of the cone.
 4. What is the formula for finding the volume of a cube?
Ans. The formula for finding the volume of a cube is given by s^3, where s represents the length of any side of the cube.
 5. How do you calculate the curved surface area of a cone?
Ans. The curved surface area of a cone can be calculated using the formula πrl, where r is the radius of the base and l is the slant height of the cone.

## NCERT Textbooks (Class 6 to Class 12)

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