JOINT VARIATIONS & PART VARIATIONS Lam Bo Jun (9) Lam Bo Xiang (10) Leong Zhao Hong (11)

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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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y kx

ky

a

kxy

a

kaxy

ax

ky

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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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• 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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• 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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Fixed profit

Selling Price

Initial price of

good

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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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2

2142

37537

5

k

k

k

kwxy

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• 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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• 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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• 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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• Sub. in the values given:

kP

v

40500

5025000 40

25000625

40

k

k

k

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• 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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• 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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5.1=30

45=

45=30

10+30=55

+=

k

k

k

pkIS

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• 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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• 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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• From the question:

20

402550

540

5030

5

k

k

kP

kDU

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• 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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• 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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• From question,

2/1.0

0.220

205.05.2

sma

a

a

atuv

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• 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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• 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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)(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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• 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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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

Documents

JOINT VARIATIONS & PART VARIATIONS Lam Bo Jun (9) Lam Bo Xiang (10) Leong Zhao Hong (11)