#### 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

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

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

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
r2 = 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

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

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

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

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.]
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

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

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.]
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

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

Step: 1
[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 × 13πr2h4r2 = 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

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
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
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