EXAMPLES ON VOLUME OF SPHERE AND HEMISPHERE

Example 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)

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

Example 2 :

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

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.

Example 3 :

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

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

Example 4 :

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

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.

Example 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.

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

7: 143

 7 ⋅ 7 : 14  14  14

1 : 8

So, the required ratio is 1 : 8.

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