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61. The annual rainfalll at a place is 43 cm. Find the weight in metric tonnes of the annual rainfall there on a hectare of land, taking the weight of water to be 1 metric tonne for 1 cubic metre.
  A.   3300 metric tonnes.
  B.   4300 metric tonnes.
  C.   4200 metric tonnes.
  D.   4400 metric tonnes.
     
   
View Answer

Shortcut:
To find the volume of rain water at a place if the annual rainfall of that place is given.
Vol. of rain water = Height (or level) of water (i.e., Annual rainfall) x Base area (i.e area of the place)
Here, level = 43/100 m, Base area = 10000 sq m
Vol. of rain water = Height (or level) of water x Base area
Vol. of water =
43 / 100
x 10000 = 4300 cubic metre
(As 1 hectare = 10,000 sq m)
∴ weight of water = 4300 x 1 = 4300 metric tonnes.

62. A rectangular tank is 50 metres long and 2 metres deep. If 1000 cubic metres of water be drawn fof the tank, the level of the water in the tank goes down by 2 metres. How many cubic metres of water can the tank hold? and also find the breadth of the tank.
  A.  Volume:16500 cubic metres,Breadth: 20 metres
  B.  Volume:14500 cubic metres,Breadth: 15 metres
  C.  Volume:14500 cubic metres,Breadth: 10 metres
  D.  Volume:15500 cubic metres,Breadth: 10 metres
     
   
View Answer

Shortcut:
A rectangular tank is 'l' metres long and 'h' metres deep. If 'z' cubic metres of water be drawn off the tank, the level of the water in the tank goes down by 'd' metres, then the amount of water (in cubic metres) the tank can hold is given by
z x h / d
cubic metres and the breadth of the tank is
z / l x d

Here, z = 1000, h = 29, d = 2, l = 50
Using these values in the shortcut, we get:
Vol. of tank =
1000 x 29 / 2
= 14500 cubic metre
∴ Breadth of the tank =
1000 / 50 x 2
= 10 metres.

63. A cubic metre of copper weighing 9000 kilograms is rolled into a square bar 9 metres long. An exact cube is cut off from the bar. How much does it weigh?
  A.  323.3 kg
  B.  353.3 kg
  C.  320.3 kg
  D.  333.3 kg
     
   
View Answer

Shortcut:
'z' cubic metres of copper weighing 'y' kg is rolled into a square bar l metres long. An exact cube is cut off from the bar. Weight of the cube is given by [√(
z / l
)]3 x y
Weight of cube = [√(
Vol. of original solid / length of the solid
)]3 x y Here, z = 1, l = 9, y = 9000
Using these values in the shortcut, we get:
Weight of cube = [√(
1 / 9
)]3 x 9000
=
9000 / 27
= 333.3 kg

64. Three cubes of metal whose edges are 3cm, 4cm and 5cm respectively are melted and formed into a single cube> if there be no loss of metal in the process find the side of the new cube.
  A.  6 cm
  B.  8 cm
  C.  4 cm
  D.  9 cm
     
   
View Answer

Shortcut:
When many cubes integrate into one cube, the side of the new cube is given by
Side = ∛(Sum of cubes of sides of all the cubes)
Here, s1 = 3, s2 = 4, s3 = 5
Using these values in the shortcut, we get:
Side = ∛(33 + 43 + 53)
= ∛(9 + 64 + 125) = ∛(216) = 6

65. A metal sheet 27 cm long, 8 cm broad and 1 cm thick is melted into a cube. The difference between the surface area of two solids will be:
  A.   296 cm2
  B.   276 cm2
  C.   286 cm2
  D.  256 cm2
     
   
View Answer

Shortcut:
Total volume of a solid does not change even when it shape changes.
∴ Old Volume = New volume
Here, Vol of new cube formed = 27 x 8 x 1 = 216 cubic cm
Edge of this cube = (216)1/3 = (6 x 6 x 6)1/3 = 6 cm
∴ Surface area of this cube = 6a2 = 6 x 6 x 6 = 216 sq cm.
∴ Surface area of given cuboid = 2(27 x 8 + 8 x 1 + 27 x 1) = 502 sq cm.
∴ Difference between the surface areas = 502 − 216 = 286 sq cm.

66. A cube of sides 3 cm is melted and smaller cubes of sides 1 cm each are formed. How many such cubes are possible?
  A.  37
  B.  27
  C.  17
  D.  47
     
   
View Answer

Shortcut:
To find the number of possible cubes when disintegration of a cube into identical cubes.
No. of cubes = (
Original length of side / New length of the side
)3
Here, Original length of side = 3, New length of side = 1
No. of cubes = (
3 / 1
)3 = 27

67. A hollow cylindrical tube open at both ends is made of iron 2 cm thick. If the internal diameter be 50 cm and the length of the tube be 140 cm, find the volume of iron in it.
  A.   44760 cu cm
  B.   45560 cu cm
  C.   45260 cu cm
  D.   45760 cu cm
     
   
View Answer

Shortcut:
A hollow cylindrical tube open at both ends is made of a thick metal. If the internal diameter or radius and length of the tube are given, then
Volume of metal = [π x height x (2 x Internal radius + thickness) x thickness] cubic units.
Here, height = 140, radius = 50/2 = 25, thickness = 2
Using these values in the shortcut, we get:
Volume of metal = [
22 / 7
x 140 x (2 x 25 + 2) x 2]
= [
22 / 7
x 140 x 52 x 2] = 22 x 20 x 52 x 2 = 45760 cubic cm

68. A hollow cylindrical tube open at both ends is made of iron. If the external and internal radius of the tube are 25 cm and 23 cm respectively, find the volume of iron in it.
  A.  41240 cu cm
  B.  42440 cu cm
  C.  42240 cu cm
  D.  43240 cu cm
     
   
View Answer

Shortcut:
A hollow cylindrical tube open at both ends is made of a thick metal. If the internal and external diameter or radius of the tube are given, then
Volume of metal = [π x height x {(External radius)2 − (Internal radius)2}] cubic units.
Here, height = 140, External radius = 25, Internal radius = 23
Using these values in the shortcut, we get:
Volume of metal = [π x height x {(External radius)2 − (Internal radius)2}] cubic units.
Volume of metal = [
22 / 7
x 140 x {(25)2 − (23)2}]
= [
22 / 7
x 140 x {625 − 529}]
= [
22 / 7
x 140 x 96] = 22 x 20 x 96 = 42240 cubic cm

69. A hollow cylindrical tube open at both ends is made of iron 2 cm thickternal diameter be 50 cm and the length of the tube be 140 cm, find the volume of iron in it.
  A.   42240 cu cm
  B.   43240 cu cm
  C.   43240 cu cm
  D.   44240 cu cm
     
   
View Answer

Shortcut:
A hollow cylindrical tube open at both ends is made of a thick metal. If the external diameter or radius and length of the tube are given, then
Volume of metal = [π x height x (2 x outer radius − thickness) x thickness] cubic units.
Here, height = 140, External radius = 50/2 = 25, Thickness = 2
Using these values in the shortcut, we get:
Volume of metal = [
22 / 7
x 140 x (2 x 25 − 2) x 2}]
= [
22 / 7
x 140 x (50 − 2) x 2}]
= [
22 / 7
x 140 x 48 x 2] = 22 x 20 x 48 x 2 = 42240 cubic cm

70. A rectangular sheet with dimesion 22m x 10 m is rolled into a cylinder so that the smaller side becomes the height of the cylinder. What is the voume of the cylinder so formed?
  A.   345 cu m
  B.   385 cu m
  C.   365 cu m
  D.   395 cu m
     
   
View Answer

Shortcut:
If a rectangular sheet is rolled into a cylinder so that the one side becomes the height of the cylinder then Volume of the cylinder =
height x (other side of the sheet)2 / 4π

Here, height = 10, other side = 22
Using these values in the shortcut, we get:
Volume of the cylinder =
10 x (22)2 / 4 x 22/7

=
10 x 22 x 22 x 7 / 4 x 22
= 5 x 11 x 7 = 385 cubic metre

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