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More examples on VARIATIONS
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Question 1
• The volume of a cylinder is directly
proportional to the square of the radius of
the base and the height
• Find the value of the volume when the
radius is 7cm and the height is (30/π)cm
0011 0010 1010 1101 0001 0100 1011
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Solution for Question 1
0011 0010 1010 1101 0001 0100 1011
• From question,
V = k r2 h
• Therefore k = π
V  r h
2
 30 
V   49  
 
3
V  1470cm
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Question 2 --Pressure
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• Pressure varies directly with
temperature and indirectly with
volume.
• The internal pressure of a tank is
500Pa when the temperature is
40°C and its volume is 50cm3.
• What is its internal pressure when
the temperature drops to 30°C and
its volume is increased to 70cm3?
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Solution Question 2
0011 0010 1010 1101 0001 0100 1011
• From the question,
k
P
v
• Sub. in the values given:
40k
500 
50
25000  40k
25000
k
 625
40
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• Now that we know the constant is 625, we
0011 0010 1010 1101 0001 0100 1011
sub. in the value given in the question again:
30  625
30k 
P
70
70
18750

70
 267.86
 268
(3s.f.)
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Question 3
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• The selling price of a certain good set by the shop owner is
directly correlated to the good’s initial buying price.
• On top of that, there is a fixed profit of $10 for every item
sold.
• The selling price of a book is $55 when its
initial price is $30.
• What’s the initial price of a telephone if it’s sold at
$190?
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Solution for Question 3
• From the question,
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S = kI + p
55 = 30k + 10
30k = 45
45
k=
= 1.5
30
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•Now that we know that the constant is 1.5, we sub. the values of the question in:
190  1.5  I  10
180  1.5 I
I Tel 
180
 $120
1 .5
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Question 4
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• PizzaHub delights its customers with its cheap and
delicious pizzas
• Suppose the average happiness of the customers (in utils)
vary directly with the diameter of the pizza (in cm) and
inversely with the price of the pizza (in S$), and has a base
of 5 utils
• The customers have an average of 30 utils when the pizza
diameter is 50cm and the price of the pizza is S$40
• How happy will the customers be when the diameter of the
pizza becomes 30cm but price rises to S$60 (for the same
flavour of pizza) due to inflation?
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Solution for Question 4
kD
P
50k
30 
5
40
50k  25  40
U 0100
 1011
5
0011 0010 1010 1101 0001
k  20
Subst. k = 20, D and P to get U
kD
5
P
20  30
U 
5
60
U 6
U 
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Question 5
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• A car travels at a constant velocity of 0.5 m/s
• It accelerates constantly to 2.5 m/s after 20s
• Given v = u + at, where v is the final
velocity, u is the initial velocity, a is
acceleration, t is time taken, find a
• Hence find the final velocity of the car after
another 20s, given constant acceleration
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Solution for Question 5
0011 0010 1010 1101 0001 0100 1011
• From question,
v  u  at
2.5  0.5  a 20 
20 a  2.0
a  0 .1
m / s2
Now we know that a = 0.1 m/s
After another 20s, u = 2.5 m/s.
v  u  at
v  2.5  0.1 20 
v  4.5 m / s
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Question 6
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• Under certain conditions, the thrust T of a
propeller varies jointly as the fourth power of its
diameter d and the square of the number n of
revolutions per second
• Show that if n is doubled, and d is halved, the
thrust T decreases by 75%
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Solution for Question 6
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4 2
T  kd n
4
d
2
T2  k   2n 
 2
 1 4 2
 k  d  4n
 16 
1 4 2
 kd n
4
75  4 2

 1 
 kd n ( proven )
 100 
 
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Question 7
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• The number of hours h that it takes m men
to assemble x machines varies directly as
the number of machines and inversely as the
number of men.
• If four men can assemble 12 machines in
four hours, how many men are needed to
assemble 36 machines in eight hours?
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Solution for Question 7
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kx
h
m
m1  4, x1  12, h1  4
12k
4
4
4
k
3
4x
h 
3m
x 2  36, h2  8
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4( 36)
8
3m
m  6
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Bibliography
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• http://www.regentsprep.org/Regents/math/algtrig/ATE7/Inverse%
20Variation.htm
• http://www.regentsprep.org/Regents/math/algtrig/ATE7/variation
%20practice%202.htm
• http://www.onlinemathlearning.com/joint-variation.html
• http://www.hci.sg/~angcc/Sec3Online/independentstudies.html
• http://www.purplemath.com/modules/variatn.htm
• http://www.purplemath.com/modules/variatn2.htm
• http://www.purplemath.com/modules/variatn3.htm
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