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41. Find the volume of a hemisphere of radius 21cm.
  A.   19404 cm3
  B.  19304 cm3
  C.  19504 cm3
  D.  19204 cm3
     
   
View Answer

Shortcut:
If the radius of a sphere is 'r' units, then the volume of a hemisphere is
(
2 / 3
πr3
)
cubic units. If diameter is given, then volume of a hemisphere is given by
(
π / 12
D3
)
cubic units.
where, D is diameter of the sphere.
Here, r = 21
Using this value in the shortcut, we get:
Vol of hemisphere =
(
2 / 3
x
22 / 7
x (21)3
)

=
2 x 22 x 21 x 21 x 21 / 3 x 7
= 19404
Hence, the volume of hemisphere is 19404 cubic cm.

42. Find the curved surface area of a hemishphere of radius 21 cm.
  A.  2778cm 2
  B.  2772 cm 2
  C.  2768 cm 2
  D.  2770 cm 2
     
   
View Answer

Shortcut:
If the radius of a sphere is 'r' units, then the curved surface area of a hemisphere is 2πr2 sq units. if in place of radius, diameter is given, then the curved surface area of the hemisphere is given by
(
π / 2
D2
)
sq units.
where, D is diameter of the sphere.
Here, r = 21
Using this value in the shortcut, we get:
curved surface area of hemisphere = 2 x
22 / 7
x (21)2
=
2 x 22 x 21 x 21 / 7
= 2772
Hence, the curved surface area of hemisphere is 2772 sq cm.

43. Find the total surface area of a hemisphere of radius 21 cm.
  A.  4152 cm 2
  B.  4158 cm 2
  C.  4162 cm 2
  D.  4160 cm 2
     
   
View Answer

Shortcut:
If the radius of a sphere is 'r' units, then the whole surface area of a hemisphere is 3πr2 sq units. If in place of radius, diameter is given, then the whole surface area of the hemisphere is given by
(
3 / 4
πD2
)
sq units.
where, D is diameter of the sphere.
Here, r = 21
Using this value in the shortcut, we get:
Whole/Total surface area of hemisphere = 3 x
22 / 7
x (21)2
=
3 x 22 x 21 x 21 / 7
= 4158
Hence, the total surface area of hemisphere is 4158 sq cm.

44. Radius of the base of a right circular cone is 3 cm and height of the cone is 4 cm.Find the slant height of the cone.
  A.  5 cm
  B.  2 cm
  C.  3 cm
  D.  7 cm
     
   
View Answer

Shortcut:
To find the slant height of the right circular cone if radius of its base and height of the cone are given. Slant height (l) = [√(h2 + r2] units.

Where, h = height and r = radius of the base.
Here, r = 3, h = 4
Using this value in the shortcut, we get:
Slant height (l) = [√(42 + 32)]
= [√(16 + 9)] = √(25) = 5
Hence, the slant height of the cone is 5 cm.

45. The diameter of the base of a right circular cone is 6 cm and its perpendicular height is 3√3. Find the slant height of the cone.
  A.  5 cm
  B.  6 cm
  C.  3 cm
  D.  7 cm
     
   
View Answer

Shortcut:
To find the slant height of the right circular cone if radius of its base and height of the cone are given. Slant height (l) = [√(h2 + r2] units. Where, h = height and r = radius of the base.
Here, D = 6 or r =
D / 2
=
6 / 2
= 3
r = 3, h = 3√3
Using this value in the shortcut, we get:
Slant height (l) = [√(3√32 + 32)]
= [√(27 + 9)] = √(36) = 6
Hence, the slant height of the cone is 6 cm.

46. Radius of the base of a right circular cone is 7 cm the height of the cone is 3 cm.Find the volume of the cone.
  A.  154 cm3
  B.  160 cm3
  C.  152 cm3
  D.  151 cm3
     
   
View Answer

Shortcut:
To find the volume of the right circular cone, if radius of the base and the height of the cone is given. Volume of the cone =
1 / 3
πr2h units.
Here, r = 7, h = 3
Using this value in the shortcut, we get:
Volume of the cone =
1 / 3
x
22 / 7
x 7 x 7 x 3
=
22 x 7 x 7 x 3 / 3 x 7
= 22 x 7 = 154 Hence, the volume of the cone is 154 cubic cm.

47. If the height of a cone is increased by 100 % , then its volume is increased by :
  A.  85%
  B.  90%
  C.  105%
  D.  100%
     
   
View Answer

Shortcut:
To find the volume of the right circular cone, if radius of the base and the height of the cone is given. Volume of the cone =
1 / 3
πr2h units.
Here, Let the height be h and radius be r. New height = 2h
Change in Volume =
1/3πr2(2h) − r21/3πr2h / 1/3πr2h
x 100 = 100%
Hence, the volume of the cone will increase by 100%.

48. If a right circular cone of vertical height 24 cm has a volume of 1232 cm 3 , then the area of its curved surface in cm2 is :
  A.  580 cm2
  B.  450 cm2
  C.  550 cm2
  D.  650 cm2
     
   
View Answer

Shortcut:
To find the volume of the right circular cone, if radius of the base and the height of the cone is given. Volume of the cone =
1 / 3
πr2h units.
Here, h = 24, vol = 1232
Volume of the cone =
1 / 3
x
22 / 7
x r2h.
1232 =
22 x 24 / 3 x 7
x r2
r2 =
1232 x 3 x 7 / 22 x 24

r2 = 7 x 7
r = 7
Slant height = √[(24)2 + (7)2] = 25 cm
∴ curved surface area =
22 / 7
x 7 x 25 = 550 sq cm

49. A right cylinderical vessel is full with water.How many right cones having same diameter and height as those of right cylinder will be needed to store the water?
  A.  5
  B.  3
  C.  1
  D.  2
     
   
View Answer

Shortcut:
To find the volume of the right circular cone, if radius of the base and the height of the cone is given. Volume of the cone =
1 / 3
πr2h units.
Here, Volume of 1 cylinder = πr2h
Volume of 1 cone =
1 / 3
πr2h
No. of cones =
πr2h / 1/3πr2h
= 3
Hence, 3 right cones will be needed to store the water.

50. A cylinderical piece of metal of radius 2 cm and height 6 cm is shaped into a cone of same radius. The height of cone is :
  A.  18 cm
  B.  21 cm
  C.  25 cm
  D.  15 cm
     
   
View Answer

Shortcut:
To find the volume of the right circular cone, if radius of the base and the height of the cone is given. Volume of the cone =
1 / 3
πr2h units.
Here, r = 2, h (cone) = ?, h (cylinder) = 6
Using these values in the shortcut, we get:
Volume of a cone =
1 / 3
π(2)2h
Vol. of cylinder = πr2h
As per question:
Vol. of cone = Vol. of cylinder
1 / 3
π 2 x 2 x h = π2 x 2 x 6
∴ h = 18
Hence, the height of the cone is 18 cm.

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