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121. The radius of a right circular cylinder is decreased by 10% but its height is increased by 15%. What is the percentage change in its volume?
  A.  Increase in volume 8.85%
  B.  Decrease in volume 6.85%
  C.  Decrease in volume 8.85%
  D.  Increase in volume 6.85%
     
   
View Answer

Shortcut:
If the radius of a right circular cylinder is changed by a% and height is changed by b%, then volume changes by
[
2a + b +
a2 + 2ab / 100
+
a2b / 1002
]
percent.
Note:
1: This theorem also holds good for right-circular cones.
2: We have used the word 'change' in place of increase or decrease. By this we conclude that if there is increase use +ve value and if there is decrease then use −ve value. If we get the answer +ve or −ve then there is respectively increase or decrease.

Here, a = −10 [− sign represents decrease],
b = 15
Using these values in the shortcut, we get:
Change in volume =
[
2(−10) + 15 +
(−10)2 + 2(−10)(15) / 100
+
(−10)2b / 1002
]
percent
=
[
−20 + 15 +
100 −300 / 100
+
10 x 10 x 15 / 100 x 100
]
percent
=
[
−5 +
−200 / 100
+
15 / 100
]
percent
= [−5 −2 + 0.15]percent
= [−7 + 0.15]% = −6.85%
Hence, the volume will decrease by 6.85%.

122. Each of the radius and the height of a right circular cylinder is both increased by 10%. Find the percentage by which the volume increases.
  A.  28.1%
  B.  35.1%
  C.  33.1%
  D.  23.1%
     
   
View Answer

Shortcut:
If the height and radius of a right circular cylinder both changes by a%, then volume changes by
[
3a +
3a2 / 100
+
a3 / 1002
]
percent.
Note:
1: This theorem also holds good for right-circular cones.
2: We have used the word 'change' in place of increase or decrease. By this we conclude that if there is increase use +ve value and if there is decrease then use −ve value. If we get the answer +ve or −ve then there is respectively increase or decrease.

Here, a = 10
Using these values in the shortcut, we get:
Change in volume =
[
3a +
3a2 / 100
+
a3 / 1002
]
percent.
=
[
3 x 10 +
3(10)2 / 100
+
(10)3 / 1002
]
percent.
=
[
30 +
3 x 10 x 10 / 100
+
10 x 10 x 10 / 100 x 100
]
percent.
=
[
30 + 3 + 0.1
]
percent.
Hence, the volume will increase by 33.01%.

123. The radius and height of a cylinder are increased by 10% and 20% respectively. Find the per cent increase in its curved surface area.
  A.  42%
  B.  22%
  C.  34%
  D.  32%
     
   
View Answer

Shortcut:
If the radius of a right circular cylinder changes by a% and its height is changed by b%, then curved surface area changes by
[
a + b +
ab / 100
]
percent.
Note:
1: This theorem also holds good for right-circular cones as you change height to slant height.
2: We have used the word 'change' in place of increase or decrease. By this we conclude that if there is increase use +ve value and if there is decrease then use −ve value. If we get the answer +ve or −ve then there is respectively increase or decrease.

Here, a = 10, b = 20
Using these values in the shortcut, we get:
Changes in curved surface area =
[
10 + 20 +
10 x 20 / 100
]
percent.
= [10 + 20 + 2] percent = 32%
Hence, the curved surface area will increase by 32%.

124. Each of the radius and the height of a cone is increased by 20%. Then find the per cent increase in volume.
  A.  72.8%
  B.  62.8%
  C.  82.8%
  D.  74.8%
     
   
View Answer

Shortcut:
If the height and radius of a right circular cone both changes by a%, then volume changes by
[
3a +
3a2 / 100
+
a3 / 1002
]
percent.
Note:
1: This theorem also holds good for right-circular cylinders.
2: We have used the word 'change' in place of increase or decrease. By this we conclude that if there is increase use +ve value and if there is decrease then use −ve value. If we get the answer +ve or −ve then there is respectively increase or decrease.

Here, a = 20
Using these values in the shortcut, we get:
Change in volume =
[
3a +
3a2 / 100
+
a3 / 1002
]
percent.
=
[
3 x 20 +
3(20)2 / 100
+
(20)3 / 1002
]
percent.
=
[
60 +
3 x 20 x 20 / 100
+
20 x 20 x 20 / 100 x 100
]
percent.
=
[
60 + 12 + 0.8
]
percent.
Hence, the volume will increase by 72.8%.

125. Two cubes each of edge 10 cm are joined to form a single cuboid. What is the surface area of the new cuboid so formed?
  A.  1200 cm2
  B.  1400 cm2
  C.  1000 cm2
  D.  1600 cm2
     
   
View Answer

Shortcut:
If two cubes each of edge 'z' metres are joined to form a single cuboid, then the surface area of the new cuboid so formed is given by 10(z)2 sq metres.
Here, z = 10
Using this values in the shortcut, we get:
Surface area of new cuboid = 10(z)2 = 10 x 10 x 10 = 1000 sq cm

126. A circular wire of radius 42 cm is cut and bent in the form of a rectangle whose sides are in the ratio of 6 : 5. Find the smaller side of the rectangle.
  A.  55 cm
  B.  60 cm
  C.  65 cm
  D.  50 cm
     
   
View Answer

Shortcut:
If a circular wire of radius 'z' units is cut and bent in the form of a rectangle whose sides are int the ratio of m : n, then the sides of the rectangle are given by
[
πz
(
m / m + n
)]
units and
[
πz
(
n / m + n
)]
units.
Here, z = 42, m = 6, n = 5
Using these values in the shortcut, we get:
Sides =
[
42 x
22 / 7
(
6 / 6 + 5
)]

=
[
6 x 22
(
6 / 11
)]
= 6 x 2 x 6 = 72 cm.
and other side =
[
42 x
22 / 7
(
5 / 6 + 5
)]

=
[
6 x 22
(
5 / 11
)]

= 6 x 2 x 5 = 60 cm
Hence, the smaller side is 60 cm.

127. A right circular cone is exactly fitted inside a cube in such a way that athe edges of teh base of the cone are touching the edges of one of the faces of the cube and the vertex is on the opposite face of the cube. If the volume of the cube is 343 cubic cm, what spproximately is the volume of the cone?
  A.  50 cubic cm
  B.  60 cubic cm
  C.  70 cubic cm
  D.  90 cubic cm
     
   
View Answer

Shortcut:
A right circular cone is exactly fitted inside a cube in such a way that the edges of the base of the cone are touching the edges of one of the faces of the cube and the vertex is on the opposite face of the cube. If the volume of the cube is given, then the volume of the cone is given by
(
π / 12
x volume of the cube
)

Here, volume of cube = 343
Using these values in the shortcut, we get:
Vol. of cone =
(
22 / 12 x 7
x 343
)

=
(
11 / 6
x 49
)
≈ 90
Hence, the volume of cone is 90 cubic cm.

128. The radii of two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3.Calculate the ratio of their volumes and the ratio of their curved surfaces.
  A.  18 : 25 and 12 : 7
  B.  10 : 27 and 10 : 7
  C.  20 : 27 and 10 : 9
  D.  25 : 27 and 20 : 9
     
   
View Answer

Here, Let their radii be 2r, 3r and heights be 5h and 3h
Then,
V1 / V2
=
π(2r)2 x 5h / π(3r)2 x 3h
=
20 / 27
= 20 : 27
S1 / S2
=
2π x 2r x 5h / 2π x 3r x 3h
=
10 / 9
= 10 : 9

129. Sum of the length, width and depth of a cuboid is 's' and its diagonal is 'd'. Its surface area is:
  A.  s2 − d2
  B.  d2 + s2
  C.  d2 − s2
  D.  s2 + d2
     
   
View Answer

Here, l + b + h = s
√(l2 + b2 + h2) = d
Squaring both sides, we get:
(l2 + b2 + h2) = d2
(l + b + h)2 = s2
⇒ (l2 + b2 + h2) + 2(lb + bh + lh) = s2
⇒ d2 + 2(lb + bh + lh) = s2
⇒ 2(lb + bh + lh) = s2 − d2
∴ Surface area = s2 − d2
[Rule: The total surface area of a cuboid if the sum of all three sides and diagonal are given is
(Sum of all three sides)2 − (Diagonal)2]

130. A right circular cone is cut off at the middle of its height and parallel to base.Call smaller cone thus formed a and remaining partB.Then :
  A.  Both equal
  B.  Vol A < Vol B
  C.  Can't be determined
  D.  Vol B < Vol A
     
   
View Answer

Here, Let 'h' be the height of the cone and 'r' be the radius of its base.
Radius of base of A =
1 / 2
r
Height of A =
1 / 2
h
Volume of A =
[
1 / 3
π x
(
1 / 2
r
)
2 x
(
1 / 2
h
)]
=
1 / 24
πr2h
Volume of B =
(
1 / 3
πr2h −
1 / 24
πr2h
)
=
7 / 24
πr2h
Hence, we see that Volume of A < Volume of B

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