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101. If the heights and the radii of two circular cylinders are in the ratio 2 : 3 and 1 : 2 respectively. Find the ratio of their curved surface areas.
  A.  1 : 3
  B.  3 : 2
  C.  1 : 2
  D.  1 : 4
     
   
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Shortcut:
If the ratio of heights and ratio of radii of two circular cylinders are given, then the ratio of their curved surface area is given by the following result.
Ratio of curved surface area = [(Ratio of radii)(Ratio of heights)]
[Note: This formula also holds good for cones, just change to slant height instead of height.]
Here, Ratio of heights = 2 : 3
Ratio of radii = 1 : 2
Using these values in the shortcut, we get:
Ratio of curved surface area = [(1 : 2)(2 : 3)]
=
1 / 2
x
2 / 3
=
1 / 3
= 1 : 3

102. If the radii and the cured surface areas of two circular cylinders are in the ratio 3 : 5 and 6 : 7 respectively. Find the ratio of their heights.
  A.  15 : 4
  B.  10 : 7
  C.  12 : 7
  D.  10 : 9
     
   
View Answer

Shortcut:
If the ratio of radii and ratio of curved surface area of two circular cylinders are given, then the ratio of their heights is given by the following result.
Ratio of heights = [(Ratio of curved surface area)(Inverse ratio of radii)]
[Note: This formula also holds good for right circular cones, just change to slant height instead of height.]
Here, Ratio of radii = 3 : 5
Ratio of curved surface area = 6 : 7
Using these values in the shortcut, we get:
Ratio of heights = [(6 : 7)
(
1 / 3
:
1 / 5
)]

= (6 : 7)(5 : 3) =
6 / 7
x
5 / 3
=
30 / 21
= 10 : 7

103. If the heights and the curved surface areas of two circular cylinders are in the ratio 1 : 3 and 4 : 5 respectively. Find the ratio of their radii.
  A.  5 : 14
  B.  12 : 5
  C.  10 : 9
  D.  12 : 7
     
   
View Answer

Shortcut:
If the ratio of heights and ratio of curved surface area of two circular cylinders are given, then the ratio of their radii is given by the following result.
Ratio of radii = [(Ratio of curved surface area)(Inverse ratio of heights)]
[Note: This formula also holds good for cones, just change to slant height instead of height.]
Here, Ratio of heights = 1 : 3
Ratio of curved surface area = 4 : 5
Using these values in the shortcut, we get:
Ratio of radii =
[
(4 : 5)
(
1 :
1 / 3
)]

= (4 : 5)(3 : 1) =
4 / 5
x
3 / 1

=
12 / 5
= 12 : 5

104. Two centimetre of rain has fallen on a square km of land. Assuming that 50% of the raindrops could have been collected and contained in a pool having a 100 m x 10 m base, by what level would the water level in the pool have increased?
  A.  10 m
  B.  12 m
  C.  16 m
  D.  15 m
     
   
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Shortcut:
'z' units of rain has fallen on a 'y' square units of land. Assuming that R% of the raindrops could have been collected and contained in a pool having a z1 units x y1 units base, the level, by which the water level in the pool would have increased, is
R / 100
x
zy / z1y1
units.
Here, R = 50, z = 2 cm, y = 1 km2
z1 = 100 m, y1 = 10 m
z = 0.02 m, y = (1000)2 m
Using these values in the shortcut, we get:
Required answer =
50 / 100
x
0.02 x 1000 x 1000 / 100 x 10
= 10 m

105. If the radius of a cylinder is doubled and the height is halved, what is the ratio between the new volume and the previous volume?
  A.  1 : 3
  B.  2 : 1
  C.  1 : 2
  D.  3 : 1
     
   
View Answer

Shortcut:
If the radius of a cylinder becomes 'z' times and the height becomes 'y' times, then the ratio between the new volume and the previous volume is given by
Required ratio = z2y
Here, z = 2, y =
1 / 2

Using these values in the shortcut, we get:
Required ratio = (2)2 x
1 / 2
= 2 : 1

106. If the radius of a cylinder is doubled and the height is halved, what is the ratio between the new curved surface area and the previous curved surface area of the cylinder.
  A.  1 : 1
  B.  2 : 3
  C.  1 : 2
  D.  2 : 1
     
   
View Answer

Shortcut:
If the radius of a circular cylinder becomes 'z' times and the height becomes 'y' times, then the ratio between the new curved surface area and the previous curved surface area is given by
Required ratio = zy
Here, z = 2, y =
1 / 2

Using these values in the shortcut, we get:
Required ratio = 2 x
1 / 2
= 1 : 1

107. A well of 11.2 m diameter is dug 8 m deep. the earth taken out has been spread all round it to a width of 7 m to form a circular embankment. Find the height of this embankment.
  A.  1.55 metres.
  B.  1.75 metres.
  C.  1.85 metres.
  D.  1.97 metres.
     
   
View Answer

Shortcut:
A well of 'D' m diameter or radius 'r' metre (here, r = D/2) is dug 'h' m deep. If the earth taken out has been spread all round it to a width of 'w' m to form a circular embankment, then the height of this embankment is given by:
Required height =
r2h / w(w + D)
metres.
Here, D = 11.2, h = 8, w = 7
r = D/2 = 11.2/2 = 5.6
Using these values in the shortcut, we get:
Required height =
(5.6)r2 x 8 / 7(7 + 11.2)

=
5.6 x 5.6 x 8 / 7 x 18.2

=
250.88 / 127.4
= 1.97 metres

108. A right-angled triangle having base 3 metres and height equal to 4 meters, is turned around the height. Find the volume of the cone thus formed. Also find the surface area.
  A.  Volume: 15π cubic m; Surface Area : 25π sq metres
  B.  Volume: 14π cubic m; Surface Area : 18π sq metres
  C.  Volume: 12π cubic m; Surface Area : 15π sq metres
  D.  Volume: 10π cubic m; Surface Area : 20π sq metres
     
   
View Answer

Shortcut:
A right-angled triangle having base 'z' metres and height equal to 'y' metres is turned around the height, a right circular cone is formed. Then,
(i) the volume of the cone =
π / 3
z2 x y cubic metres.
(ii) the surface area of the cone = πz√(z2 + y2)] sq metres.
Here, z = 3, h = 4
Using these values in the shortcut, we get:
The volume of the cone =
π / 3
x 32 x 4
=
π / 3
x 9 x 4 = π x 3 x 4 = 12π cubic metres.

The surface area of the cone = [π x 3 x √(32 + 42)]
= [π x 3 x √(9 + 16)]
= [π x 3 x √(25)]
= π x 3 x 5 = 15π sq metres

109. A right-angled triangle having base 3 metres and height equal to 4 metres, is turned around the base. Find the volume of the cone thus formed. Also find the surface area.
  A.  Volume:12π cu m; Surface area:16π sq m
  B.  Volume:16π cu m; Surface area:20π sq m
  C.  Volume:14π cu m; Surface area:24π sq m
  D.  Volume:18π cu m; Surface area:22π sq m
     
   
View Answer

Shortcut:
A right-angled triangle having base 'z' metres and height equal to 'y' metres is turned around the base, a right circular cone is formed. Then,
(i) the volume of the cone =
π / 3
z x y2 cubic metres.
(ii) the surface area of the cone = πy√(z2 + y2)] sq metres.
Here, z = 3, h = 4
Using these values in the shortcut, we get:
The volume of the cone =
π / 3
x 3 x 4 x 4 = 16π cubic metres.
The surface area of the cone = π x 4 √(32 + 42)]
= π x 4 √(9 + 16)]
= π x 4 √(25)]
= π x 4 x 5 = 20π sq metres

110. The length, breadth and height of a cuboid are increased by 5%, 10% and 15% respectively.Find the percentage increase in its volume.
  A.  32.825%
  B.  38.325%
  C.  35.225%
  D.  30.525%
     
   
View Answer

Shortcut:
If length, breadth and height of a cuboid is increased by a%, b% and c% respectively, then
Increase in volume =
[
a + b + c +
ab + bc + ca / 100
+
abc / (100)2
]
%.
Here, a = 5, b = 10, c = 15
Using these values in the shortcut, we get:
Increase in volume =
[
5 + 10 + 15 +
5 x 10 + 10 x 15 + 15 x 5 / 100
+
5 x 10 x 15 / (100)2
]
%
=
[
30 +
50 + 150 + 75 / 100
+
750 / (100)2
]
%
=
[
30 +
275 / 100
+
750 / 100 x 100
]
%
= 30 + 2.75 + 0.075 = 32.825 %

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