JOINT VARIATIONS & PART VARIATIONS Lam Bo Jun (9) Lam Bo Xiang (10) Leong Zhao Hong (11)
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Transcript of JOINT VARIATIONS & PART VARIATIONS Lam Bo Jun (9) Lam Bo Xiang (10) Leong Zhao Hong (11)
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0011 0010 1010 1101 0001 0100 1011JOINT VARIATIONSJOINT VARIATIONS&&
PART VARIATIONSPART VARIATIONS
Lam Bo Jun (9)Lam Bo Xiang (10)
Leong Zhao Hong (11)
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Recap:Recap:• Direct Variation
y x
y kx
1y
xk
yx
• Indirect Variation
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Steps to do Variations QuestionsSteps to do Variations Questions
1. Write the equation
2. Substitute given valuesFind the constant of variation, k
3. Solve the values
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• Physics equations are included to facilitate revision for tomorrow’s test
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JOINT VARIATIONSJOINT VARIATIONS
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Joint variationsJoint variations
• Different combinations of direct and indirect variations
• Used when a certain variable is – affected by two or more other
variables, – directly and/or indirectly.
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Joint variationsJoint variations
y kx
ky
a
kxy
a
kaxy
ax
ky
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Joint variationsJoint variations
Distance
Speed
Time
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Joint variations (Example)Joint variations (Example)
• y varies directly with x and indirectly with a.
• y = 10 when x = 5 and a = 3
• What is its value of y when x = 13 and a = 5?
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Joint variations (Example)Joint variations (Example)
• From the question,
• Sub. in the values given: y = 10 when x = 5 and a = 3
a
kxy
6
3053
510
k
k
k
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Joint variations (Example)Joint variations (Example)
• Now that we know the k = 6, we sub. in the value given in the question again: x = 13, a = 5
6.155
136
6
y
y
a
xy
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Some real life examples…Some real life examples…• The volume of a cylinder
V = π r2 h
• The power of a heater (in Watts)
• Specific Heat Capacity
m
Qc
t
EP
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PART VARIATIONSPART VARIATIONS
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Part variationsPart variations
• Contains direct or indirect variation or even both
• Used when an extra constant is added to the affected variable, neither directly nor indirectly affecting it
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Part variationsPart variations
Fixed profit
Selling Price
Initial price of
good
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Part variationsPart variations
y kx C ky C
x
kxy C
a
Ckaxy k
y Cax
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Part variations (Example)Part variations (Example)
• The value of y + 5 is directly related to w and x
• When w = 7 and x = 3, y = 37
• Find the value of y when w = 13 and x = 21
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Part variations (Example)Part variations (Example)
• From the question,
2
2142
37537
5
k
k
k
kwxy
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Part variations (Example)Part variations (Example)
• Now that we know that the constant is 2, we sub. the values from the question in:
541
5546
521132
5
y
kwxy
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Some real life examples…Some real life examples…
• Kinematics v = u + at v2 = u2 + 2as s = ut + ½ at2
• Dynamics F = m [(v – u) / t]
And some PHYSICS revision
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PRACTICE EXERCISESPRACTICE EXERCISES
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VARIATIONS Q1VARIATIONS Q1CYLINDER
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Variations (Question 1)Variations (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
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Variations (Question 1)Variations (Question 1)
• From question,V = k r2 h
• Therefore k = π
3
2
1470
3049
cmV
V
hrV
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VARIATIONS Q2VARIATIONS Q2PRESSURE
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Variations (Question 2)Variations (Question 2)
• 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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Variations (Question 2)Variations (Question 2)
• From the question,
• Sub. in the values given:
kP
v
40500
5025000 40
25000625
40
k
k
k
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Variations (Question 2)Variations (Question 2)
• Now that we know the constant is 625, we sub. in the value given in the question again:
30
70
kP
30 625
7018750
70267.86
268
(3s.f.)Pa
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VARIATIONS Q3VARIATIONS Q3PRICE
http://news-lib
raries.mit.ed
u/blog/wp-co
ntent/uploads
/2008/01/mo
ney.jpg
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Variations (Question 3)Variations (Question 3)
• 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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Variations (Question 3)Variations (Question 3)
• From the question,
5.1=30
45=
45=30
10+30=55
+=
k
k
k
pkIS
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Variations (Question 3)Variations (Question 3)
• Now that we know that the constant is 1.5, we sub. the values of the question in:
120$5.1
180
5.1180
105.1190
TelI
I
I
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VARIATIONS Q4VARIATIONS Q4PIZZA
http://www.resch-frisch.com/images/products/Pizza-Capricciosa.jpg
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Variations (Question 4)Variations (Question 4)
• PizzaHub delights its customers with its cheap and delicious pizzas
• Suppose the average happiness of the customers (in utils) vary directly with the diameter of the pizza (in cm) and inversely with the price of the pizza (in S$), 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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Variations (Question 4)Variations (Question 4)
• From the question:
20
402550
540
5030
5
k
k
kP
kDU
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Variations (Question 4)Variations (Question 4)
• Now we know that k = 20,Sub D and P to get U
6
560
3020
5
U
U
P
kDU
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VARIATIONS Q5VARIATIONS Q5VELOCITY
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Variations (Question 5)Variations (Question 5)
• 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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Variations (Question 5)Variations (Question 5)
• From question,
2/1.0
0.220
205.05.2
sma
a
a
atuv
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Variations (Question 5)Variations (Question 5)
• Now we know that a = 0.1 m/s
• After another 20s, u = 2.5 m/s. Therefore:
smv
v
atuv
/5.4
201.05.2
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VARIATIONS Q6VARIATIONS Q6THRUST
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Variations (Question 6)Variations (Question 6)
• 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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Variations (Question 6)Variations (Question 6)
)(100
751
4
1
416
1
22
24
24
24
24
2
24
provennkd
nkd
ndk
nd
kT
nkdT
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VARIATIONS Q7VARIATIONS Q7MACHINES
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Variations (Question 7)Variations (Question 7)
• 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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Variations (Question 7)Variations (Question 7)
m
xh
k
k
hxmm
kxh
3
43
44
124
4,12,4 111
63
)36(48
8,36 22
mm
hx
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BibliographyBibliography
• 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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THANK YOUTHANK YOU