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71. A copper shphere of diameter 18 cm is drawn into a wire of diameter 4 mm. Find the length of the wire.
  A.  24300 cm
  B.  23300 cm
  C.  24500 cm
  D.  25500 cm
     
   
View Answer

Shortcut:
If a sphere of certain diameter or radius is drawn into a cylinder of certain diameter or radius, then the Length or height of the cylinder =
4 x (radius of sphere)3 / 3 x (radius of cylinder)2

Here, radius of sphere = 90, radius of cylinder = 2
Using these values in the shortcut, we get:
Length or height of the cylinder =
4 x (90)3 / 3 x (2)2

4 x 90 x 90 x 90 / 3 x 2 x 2
= 30 x 90 x 90 = 243000 = 24300 cm

72. A cylinder of radius 2 cm and height 15 cm is melted and the same mass is used to create a sphere. What will be the radius of the sphere?
  A.  ∛55
  B.  ∛45
  C.  ∛35
  D.  ∛42
     
   
View Answer

Shortcut:
A sphere is converted into a cylinder. If the lenght and the radius of the cylinder are given, then
Radius of the sphere =
3 / 4
(length of cylinder)(radius of cylinder)2
Here, radius of cylinder = 2, length of cylinder = 15
Using these values in the shortcut, we get:
Radius of the sphere = ∛[
3 / 4
(15)(2)2]
= ∛[
3 / 4
x 15 x 2 x 2]
= ∛(3 x 15) = = ∛(45)

73. A copper shpere of 36 m diameter is drawn into a cylindrical wire of lenght 7.29 km. What is the radius of wire.
  A.   1.50 m
  B.   1.75 m
  C.   1.03 m
  D.   1.15 m
     
   
View Answer

Shortcut:
If a sphere of certain diameter or radius is drawn into a cylinder of certain height or length, then
Radius of the cylinder =
4(radius of sphere)3 / 3(length of cylinder)

Here, radius of sphere = 36/2 = 18, length of cylinder = 7290
Using these values in the shortcut, we get:
Radius of the cylinder =
4(18)3 / 3 x 7290

=
4 x 18 x 18 x 18 / 3 x 7290

=
16 / 15
= 1.03 m

74. A sphere is melted to form a cylinder whose height is 4
1 / 2
times its radius. What is the ratio of radii of sphere to the cylinder?
  A.   3 : 5
  B.   2 : 5
  C.   3 : 2
  D.   5 : 2
     
   
View Answer

Shortcut:
If sphere is melted to form a cylinder whose height is 'n' times its radius, then the Ratio of radii of sphere to the cylinder = (
3 / 4
x n)1/3
Here, n = 9/2
Using these values in the shortcut, we get:
Ratio of radii of sphere to the cylinder = (
3 / 4
x 9/2)1/3
= (
3 x 9 / 4 x 2
)1/3
= (
27 / 8
)1/3
= (
3 / 2
)3 x 1/3
=
3 / 2
= 3 : 2

75. A cone whose height is half of its radius, is melted to form a sphere. Find the ratio of radius of the sphere to that of the cone.
  A.   1 : 2
  B.   1 : 3
  C.   3 : 2
  D.   1 : 5
     
   
View Answer

Shortcut:
If a cone, whose height is 'n' times of its radius, is melted to form a sphere, assuming that there is no loss of material in this process, Ratio of radius of the sphere to that of the cone = (
n / 4
)1/3
Here, n = 1/2
Using these values in the shortcut, we get:
Ratio of radii of sphere to the cylinder = (
1/2 / 4
)1/3
= (
1 / 8
)1/3
= (
1 / 2
)3 x 1/3
=
1 / 2
= 1 : 2

76. How many bulletes can be made out of a lead cylinder 28 cm high and 6 cm radius, each bullet being 1.5 cm in diameter?
  A.  1642
  B.  1872
  C.  1992
  D.  1792
     
   
View Answer

Shortcut:
When one cylinder is converted into many small spheres, then
Number of small spheres =
Vol. of cylinder / Vol. of 1 sphere

Here, r = 6, h = 28, rs = 0.75
Using these values in the shortcut, we get:
Number of small bullet =
Vol. of cylinder / Vol. of 1 bullet

=
πr2h / 4/3πrs3

=
π x 6 x 6 x 28 / 4/3 x π x 0.75 x 0.75 x 0.75
= 2 x 2 x 7 x 4 x 4 x 4 = 1792
Hence, Number of bullets is 1792.

77. A sphere of radius 5 cm has a spherical cavity of radius 3 cm. Find the volume of the spherical shell.
  A.  410
2 / 3
cm3
  B.  470
2 / 3
cm3
  C.  310
2 / 3
cm3
  D.  450
2 / 3
cm3
     
   
View Answer

Shortcut:
If a sphere of radius R units has a spherical cavity of radius 'r' units, then the Volume of the spherical shell = [
4 / 3
π(R3 − r3)]
Here, R = 5, r = 3
Using these values in the shortcut, we get:
Volume of the spherical shell = [
4 / 3
x
22 / 7
(53 − 33)]
= [
4 x 22 / 3 x 7
x (125 − 27)]
= [
4 x 22 / 3 x 7
x 98]
= [
4 x 22 / 3
x 14] = 410
2 / 3

Hence, volume of spherical shell is 410
2 / 3
cubic cm.

78. There is a cone of radius 3 metres and height 4 metres. Find the radius of the greatest sphere that can be carved out of that cone.
  A.  1.5 metres
  B.  2.5 metres
  C.  0.5 metres
  D.  3.5 metres
     
   
View Answer

Shortcut:
The radius of a greatest sphere that can be carved out of a cone of radius 'r' and height 'h' is
rh / r + l

l = slant height of cone = √(r2 + h2)
Here, h = 4, r = 3
l = √(32 + 42)
= √(9 + 16) = = √(25) = 65
Using these values in the shortcut, we get:
Radius of the greatest sphere =
3 x 4 / 3 + 5

=
3 x 4 / 8
=
3 / 2
= 1.5
Hence, the radius of the greatest sphere is 1.5 m.

79. Find the number of lead balls of diameter 1 cm each that can be made from a sphere of diameter 16 cm.
  A.  4050
  B.  4096
  C.  3096
  D.  4026
     
   
View Answer

Shortcut:
When a sphere disintegrates into many identical spheres, then the number of smaller identical spheres are given by (
bigger radius / Smaller radius
)3
Here, bigger radius = 8, Smaller radius = 0.5
Using these values in the shortcut, we get:
Required number = (
8 / 0.5
)3
= (
8 x 10 / 5
)3 = 163 = 4096

80. The curved surface areas of two spheres are in the ratio 1 : 4. Find the ratio of their volumes.
  A.  1 : 4
  B.  1 : 9
  C.  5 : 3
  D.  1 : 8
     
   
View Answer

Shortcut:
If the ratio of surface areas of the two spheres are given, then the ratio of their volumes will be obtained from the following result. (Ratio of the surface areas)3 = (Ratio of volumes)2
Here, Ratio of the surface areas = 1 : 4
Using these values in the shortcut, we get:
(1 : 4)3 = (Ratio of volumes)2
1 : 64 = (Ratio of volumes)2
(1 : 8)2 = (Ratio of volumes)2
Ratio of volumes = 1 : 8

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