VOLUME OF SPHERE AND HEMISPHERE WORKSHEET

(1)  Find the mass of 200 steel spherical ball bearings, each of which has radius 0.7 cm, given that the density of steel is 7.95 g/cm³. (Mass = Volume x Density).

(2)  The outer and inner radii of a hollow sphere are 12 cm and 10 cm. Find its volume.

(3)  The volume of a solid hemisphere is 1152 Π cu.cm. Find its curved surface area.

(4)  Find the volume of the largest right circular cone that can be cut of a cube whose edge is 14 cm.

(5)  The radius of a spherical balloon increase from 7 cm to 14 cm as air is being pumped into it. Find the ratio of volumes of the balloon in the two cases.

(6)  A ball of gold has a radius of 9cm. The density of gold is 19.3g/cm³. Work out the mass of the ball.

(7)  A hemispherical bowl has a radius of 10 cm.

(a) Calculate the volume of the bowl.

(b) A cylinder of radius 7 cm and height h cm has the same volume as the bowl. Calculate the value of h .

(8)  The total surface area of a hemisphere is given as 618 cm².

(a) Find the radius of the hemisphere.

(b) Find the volume of the hemisphere.

(c)  Find the external surface area of the hemisphere if it is hollow. Give your answers to 3 significant figures. Take π = 3.142

(9)  Given that the volume of a sphere is 5276 cm³, find its radius and surface area. Give your answers to 3 significant figures. Take π = 3.142

(10)  A hemisphere has diameter 22.4 cm. Find the volume and closed surface area. Give your answers to 3 significant figures. Take π = 3.142

(1)  Solution :

radius of spherical ball = 0.7 cm 

Volume of one spherical ball  =  (4/3) Π r³

=  (4/3) (22/7)  ⋅  0.7 ⋅  0.7 ⋅  0.7

=  4.312/3

=  1.437

Volume of 200 steel spherical ball  =  200 ⋅ 1.437

=  287.46 cm³

1 cm³ = 7.95 g

Therefore mass of 200 spherical ball bearings

=  287.46 (7.95)

=  2285.307 gram

1000 gram  =  1 kg

=  2285.307/1000

=  2.29 kg

Volume of 200 spherical balls = 2.29 kg.

(2)  Solution :

From this information we have to find the volume

Outer radius (R)  =  12 cm

Inner radius (r)  =  10 cm

Volume of hollow sphere  =  (4/3) Π (R³-r³)

=  (4/3)(22/7) (12³-10³)

=  (88/21) (1728-1000)

=  (88/21) (728)

=  (64064/21)

=  3050.67 cm³

Volume of hollow sphere is 3050.67 cm³.

(3)  Solution :

Volume of hollow sphere  =  1152 Π

(2/3) Π r³  =  1152 Π

r³  =  1152 Π (3/2Π)

r³  =  (576 x 3)/2

r³  =  1728

r  =  ∛1728

r  =  12 cm

Curved surface area  =  2Πr²

=  2Π(12)²

=  2Π(144)

=  288Π cm³

Curved surface area  =  288Π cm³

(4)  Solution :

Since it is cube length of all sides will be equal that is 14 cm. Diameter and height of cone are 14 cm.

r  =  14/2  ==>  7

h  =  14 cm

Volume of cone  =  (1/3) Π r² h 

=  (1/3) ⋅ (22/7)  ⋅ 7² ⋅ 14

=  (1/3) ⋅ 22 ⋅ 49 ⋅ 2  

=  (49 ⋅ 44)/3

=  2156/3

=  718.67 cm³

Volume of cone is 718.67 cm³.

(5)  Solution :

Let r₁ and r₂ are the radii of two spherical balloon

r₁ : r₂  =  7 : 14

Volume of one spherical balloon  =  (4/3) Π r³

(4/3) Π (7)³ :  (4/3) Π 14³

7³ : 14³

7 ⋅ 7 ⋅ 7 : 14 ⋅ 14 ⋅ 14

1 : 8

So, the required ratio is 1 : 8.

(6)  Solution :

radius = 9 cm

Density = 19.3  g/cm³

Mass = volume x density

volume of hemisphere = (2/3) Π r³

= (2/3) x 3.14 x 9³

= 1526.04 cm³

Mass = 1526.04 x 19.3

= 29452.57 g

1000 gram = 1 kg

= 29452.57/1000

= 29.45 kg

(7) Solution :

a)  

r = 10 cm

Volume of bowl = (2/3) Π r³

=  (2/3) Π (10)³

= (2/3) x 3.14 x 1000

= 2093.3 cm³

(b)  Radius = 7 cm

height = h cm

Volume of cylinder = Π r² h

= 3.14(7)² h

= 153.86 h cm³

(8)  Solution :

The total surface area of a hemisphere = 618 cm².

(a) 3 Π r²  = 618 cm²

Applying the value of π, we get

3 x 3.142 x r²  = 618

r²  = 618 / (3 x 3.142)

r²  = 65.56

r = √65.56

r = 8.09 cm

(b) volume of the hemisphere = (2/3) Π r³

= (2/3) x 3.14 x 8.09³

= 1108.36 cm³

Volume of hemisphere is  1108.36 cm³

(c)  External surface area = 2 Π r² 

= 2 x 3.14 x 8.09²

= 411.01 cm²

So, the external surface area is 411.01 cm²

(9) Solution :

Volume of a sphere = 5276 cm³

(2/3) Π r³ = 5276

(2/3) x 3.14 x r³ = 5276

r³ = (5276/3.14) x (3/2)

r³ = 2520.38

r = 13.60 cm

So, the radius of the hemisphere is 13.60 cm

(10) Solution :

diameter of hemisphere = 22.4 cm

radius (r) = 22.4/2

= 11.2 cm

Volume of hemisphere = (2/3) Π r³

= (2/3) x 3.14 x 11.2³

= 2940.98

Approximately the volume is 2941 cm³

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