Volume of a Sphere-Gr 8-Solved Examples

Solved Examples and Worksheet for Volume of a Sphere

Q1What is the base radius of the sphere, if the volume of the sphere is 256 cm³? [Use π = 3 cm.]
A. 7 cm
B. 6 cm
C. 5 cm
D. 4 cm

Solutions

Step: 1

The volume of the sphere = 256 cm³

Step: 2

Let r be the radius of the sphere.

Step: 3

The volume of the sphere V = 43πr³

Step: 4

43πr³ = 256

Step: 5

43 × 3 × r³ = 256

[Substitute π = 3.]

Step: 6

r³ = 2564

Step: 7

= 64

[Divide 256 by 4.]

Step: 8

r = (64)^1/3

Step: 9

= 4 cm

[Simplify.]

Step: 10

The base radius of the sphere is 4 cm.

Correct Answer is : 4 cm

Q2Find the maximum volume of the sphere in cubic cm that can be carved out of a cube of side 24 cm.[Take π = 3.14]
A. 7234.56 cm³
B. 13824 cm³
C. 1809.79 cm³
D. 57876.5 cm³

Solutions

Step: 1

The side of the cube is 24 cm.

[Given.]

Step: 2

The maximum radius of sphere that can be carved out of the cube = 242 = 12 cm.

Step: 3

The maximum volume of the sphere that can be carved out of the cube = 43πr³

[Formula.]

Step: 4

= 43π × 12³

Step: 5

= 7234.56 cm³

Correct Answer is : 7234.56 cm³

Q3The numerical values of the curved surface area and the volume of the hemisphere are same. What is the volume of the hemisphere?
A. 60.25 cubic units
B. 50.08 cubic units
C. 56.52 cubic units
D. 25.02 cubic units

Solutions

Step: 1

Curved surface area of the hemisphere = 2πr²
Volume of the hemisphere = 23πr³

[Formula]

Step: 2

Curved surface area of the hemisphere = volume of the hemisphere

[According to the data given]

Step: 3

2πr² = 23πr³

[Substitute]

Step: 4

r = 3units

Step: 5

Volume of hemisphere = 23π(3)³ = 56.52 cubic units.

Correct Answer is : 56.52 cubic units

Q4Three hemispheres of radius 5, 6 and 4 respectively are melted to form a sphere. What is the radius of the new sphere formed? [Take π = 3.]
A. r = 202.53 units
B. r = 200.53 units
C. r = 207.53 units
D. r = 212.53 units

Solutions

Step: 1

Volume of the sphere = 43 π r³

[Formula]

Step: 2

Volumes of three hemispheres = 23 π (5)³ + 23 π(6)³ + 23 π (4)³

[Volume of the hemisphere
= 23π (r³).]

Step: 3

As the sphere is formed by melting three hemispheres, volume of the sphere is equal to the volumes of three hemispheres.

Step: 4

43 π r³ = 23 π (5)³ + 23 π (6)³ + 23 π (4)³

Step: 5

r = 202.53 units

Correct Answer is : r = 202.53 units

Q5A cylindrical jar of radius 20 cm is filled with water upto a height of 30 cm. 15 spherical balls of radii 3 cm each are immersed in the jar. Find the new level to which water is filled in the jar. [Take π = 3.]
A. 38.35 cm
B. 33.35 cm
C. 36.35 cm
D. 31.35 cm

Solutions

Step: 1

Volume of one sphere = 43× 3 × (3³) = 108 cm³

[Volume of the sphere = 43π (r³).]

Step: 2

Volume of 15 spheres = 15 × 108 = 1620 cm³

[Multiply.]

Step: 3

Volume of water in the jar = 3 × (20²) × 30 = 36000 cm³

[Volume of the cylinder = π (r²) h.]

Step: 4

Total volume of water + balls(V) ⇒ 1620 + 36000 = 37620 cm³

[Simplify.]

Step: 5

Volume of the water in the cylinder when spherical balls are immersed = 37620 cm³

[From step 4.]

Step: 6

⇒ 3 × (20)² h = 37620 cm³

[From step 5.]

Step: 7

Height to which water is filled in the jar ⇒ 376203 × (20)² = 31.35 cm

Correct Answer is : 31.35 cm

Q6Find the sum of the volume of a cone with radius 2 cm and height 7 cm and the volume of a sphere of radius 2 cm. [Take π = 3.]
A. 64 cm³
B. 65 cm³
C. 63 cm³
D. 60 cm³

Solutions

Step: 1

Volume of the cone = 13× 3 × (2²) 7 = 28 cm³

[Volume of the cone = 13π (r²) h.]

Step: 2

Volume of the sphere = 43× 3 × (2³) = 32 cm³

[Volume of a sphere = 43π (r)³.]

Step: 3

Sum of the volumes = 28 + 32 = 60 cm³

[Simplify.]

Correct Answer is : 60 cm³

Q7An adhesive compound in liquid form is prepared in a container of hemispherical shape having a radius of 120 cm. This compound is to be packed in cylindrical bottles of radius 3 cm and height of 6 cm. How many bottles are needed if the liquid prepared exactly fills the container? [Take π = 3.]
A. 21333
B. 533
C. 42666
D. 259200

Solutions

Step: 1

Number of bottles required = total volume of the liquidvolume of one bottle

[Formula.]

Step: 2

Volume of the liquid = Volume of the hemisphere ⇒ 23× 3 × 120³ = 3456000 cm³

[Volume of the hemisphere = 23π r³.]

Step: 3

Volume of one bottle ⇒ 3 × (3)² × 6 = 162 cm³

[Volume of the cylinder = π (r)² h.]

Step: 4

Number of bottles ⇒ 3456000162 = 21333.33 (approximately)

[From steps 2 and 3.]

Correct Answer is : 21333

Q8A water tank is in the form of a hemisphere with radius 3 m. It is filled with water at a rate of 4 litres/sec. How much time will it take to fill the tank? [Take π = 3.]
A. 283 min
B. 225 min
C. 336 min
D. 292 min

Solutions

Step: 1

Radius of the tank = 3 m

[Given.]

Step: 2

Volume of the tank = 23× 3 × (3)³ × 1000 litres = 54000 litres

[Volume of the tank = 23π r³.]
[1 m³ = 1000 litres.]

Step: 3

Rate of filling the water = 4 litres/sec,

[Given.]

Step: 4

Time taken to fill the tank = Volume of the tank rate of filling

[Formula.]

Step: 5

Time taken to fill the tank = 540004
= 13500 seconds = 225 min

[Simplify.]
[1 minute = 60 seconds.]

Correct Answer is : 225 min

Q9A rectangular block of lead with dimension 40 cm × 50 cm × 60 cm is melted to mould spherical balls of 2 cm radius. How many balls are made?
[Round your answer to nearest whole number and take π = 3. ]

A. 4250
B. 3750
C. 3850
D. 4050

Solutions

Step: 1

Nuumber of spherical balls made = Total volume of leadvolume of one spherical ball

[Formula.]

Step: 2

Total volume of lead = 40 cm × 50 cm × 60 cm ⇒ 120000 cm³

[Given.]

Step: 3

Volume of one spherical ball = 43 × 3 × (2)³ ⇒ 32 cm³

[Volume of a sphere = 43π r³.]

Step: 4

Number of spherical balls = 12000032 ⇒ 3750 (approximately)

[Simplify.]

Correct Answer is : 3750

Q10An iron rod of diameter 0.5 inches was made by melting five iron spheres of radius 1.5 inches. What is the length of the iron rod in meters? [Take π = 3.]
A. 28.52 m
B. 9.14 m
C. 32.52 m
D. 18.52 m

Solutions

Step: 1

Length of the iron rod = Volume of the rod base area

[Formula.]

Step: 2

Volume of each sphere = 43× 3 × (1.5)³ ⇒ 13.50 inches³

[Volume of the sphere = 43π r³.]

Step: 3

Volume of five spheres = 5 × 13.50 ⇒ 67.50 inches³

[From step 2.]

Step: 4

Base area of the rod = 3 × (0.5)24 = 0.1875 inches²

[Base area of the rod = πd²4 .]

Step: 5

Length of the rod = 67.500.1875= 360 inches = 9.14 m

[Substitute in step1 and simplify.]
[1 inch = 0.0254 meter.]

Correct Answer is : 9.14 m

Q11A sphere and a cone have equal radii. If the volume of the sphere is double that of the cone, then the relation between height and radius of the cone is ______.
A. h = r4
B. h = 4r
C. h = 2r
D. h = r2

Solutions

Step: 1

Radius of the sphere = radius of the cone = r

[Given.]

Step: 2

Volume of the sphere = 43πr³

[Formula.]

Step: 3

Volume of the cone = 13πr²h

[Formula.]

Step: 4

Volume of the sphere = 2 × Volume of the cone

[Given.]

Step: 5

43πr³ = 2 × 13πr²h ⇒ 4r2 = h

[Substitute and simplify.]

Step: 6

Therefore, h = 2r.

Correct Answer is : h = 2r

Q12The numerical values of the surface area and the volume of the sphere are same. What is the volume of the sphere ?

A. 123.08 cubic units
B. 120.08 cubic units
C. 113.04 cubic units
D. 100.04 cubic units

Solutions

Step: 1

Surface area of the sphere = 4π r ²

Step: 2

Volume of the sphere = 43 π r³

[Formula]

Step: 3

Surface area of the sphere = volume of the sphere

[According to the data given]

Step: 4

4π r² = 43 π r³

[Substitute]

Step: 5

r = 3 units

Step: 6

Volume of sphere = 43 π (3)³ = 113.04 cubic units.

Correct Answer is : 113.04 cubic units

Solved Examples and Worksheet for Volume of a Sphere

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