5.0 Variations

8/3/2019 5.0 Variations

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CHAPTER : Variations

: Variations5

5

AState the changes in a quantity with respect to the changesin another quantity in a direct variation

B Determine whether a quantity varies directly as

another quantity

C Express a direct variation as an equation in two variables

D Find the value of a variable in a direct variation given

Sufficient information

5.1 Direct Variations


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: Variations

5

5

5.2 Inverse Variations

AState the changes in a quantity with respect to thechanges in another quantity in an inverse variation

B Determine whether a quantity varies inversely as anotherquantity

CExpress an inverse variation as an equation in twovariables

D Find the value of a variable in an inverse variation givenSufficient information


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: Variations

5

5

5.3 Joint Variations

A Represent a joint variation by using the symbol

B Express a joint variation in the form of an equation

C Find the value of a variable in a joint variation givensufficient information


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CHAPTER

A State the changes in a quantity with respect to the changesin another quantity in a direct variation

In everyday life, there are situations where an increase in the value of aquantity will cause a corresponding increase in the value of another quantity

and vice-versa.

EX

AMPLE

The table below shows the amount Ali paid for different number ofexercise books he bought

Number of exercise books bought 2 4 8 12

Amount paid (RM) 4 8 16 24

State the change in the amount Ali paid when the number of exerciseBooks he bought is(a) doubled (b) halved (c) trebled

5.1 Direct Variations

: Variations

: Variations5

5


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CHAPTER 5

: Variations5

: Variations

AState the changes in a quantity with respect to the changes

in another quantity in a direct variation

SO

LUT

IO

N

Number of exercise books bought 2 4 8 12

Amount paid (RM) 4 8 16 24

From the table, it can be seen that the more exercise books Ali buys, thegreater the amount he has to pay.

qThe amount Ali paid is doubledwhen the number of exercise books hebought is doubled

qThe amount Ali paid is halvedwhen the number of exercise books hebought is halved

qThe amount Ali paid is trebledwhen the number of exercise books hebought is trebled


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CHAPTER 5 : Variations5 : Variations

B Determine whether a quantity varies directly as another quantity

To determine whether y varies directly as x, just check whether y is ax

constant for all given pairs of (x, y).

EXAMPLE

Based on the values of x and y given in the tables below, determinewhether y varies directly as x in each case.

x 2 3 5 6 8

y 10 15 25 30 40


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SOLUT

ION x 2 3 5 6 8

y 10 15 25 30 40

yx 55

5 5 5 A constant

Hence, y varies directly as x.


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EXAMPLE

Based on the values of x and y given in the tables below, determinewhether y varies directly as x in each case.

x 2 4 6 8 12

y 5 10 15 24 30


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SOLUT

ION x 2 4 6 8 12

y 5 10 15 24 30

yx 2.52.5

2.5 3 2.5Not

a constant

Hence, y does not vary directly as x.


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If y xn, then y = kor y = kxn, where kis the constant of variation.xn

EXAMPLE

(a)Given that y varies directly as x and y = 8 when x = 12, expressy in terms of x.

(b) If p varies directly as r3 and p = 2 when r = , find the equationwhich relates p and r.


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SOLUT

ION

(a)Given y x.Then, y = kx, where k is the constant of variation

Given y = 8 when x = 12

= k( )

12

8 = k

12k = 2

3

Thus, y = 2x3

(a)Given that y varies directly as x and y = 8 when x = 12, expressy in terms of x.

8


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SOLUT

ION

(a)Given p r

3

.Then, p = kr3, where k is the constant of variation

Given p = 2 when r =

= k( )3

2 = 1k

8k = 16

Thus, p = 16r3

2

(b) If p varies directly as r3 and p = 2 when r = , find the equationwhich relates p and r.


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D Find the value of a variable in a direct variation given sufficient

information

If y varies directly as xn, then y = k or y = kxn, where k is thexn

constant of variation. Hence, given sufficient information, thevalue of x or y can be determined by using y = kxn.


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information

EXAMPLE

Given that y varies directly as x and y = 18 when x = 3, find(a) the value of y when x = 5,(b) the value of x when y = 4.

SOLUTION

Given y x y = kx


18 = k(3)k = 6Hence, y = 6x

(a) When x = 5, y = 6(5) = 30

(b) When y = 4, 4 = 6x

4 = x6

x = 23


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information

ANOTHER WAY

y1

x1

y2

x2

= (a) Let x1 = 3, y1 = 18 and x2 = 5

18 = y23 5y2 = 30

(b) Let x1 = 3, y1 = 18 and y2 = 4

18 = 4

3 x2x2 = 2

3


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A State the changes in a quantity with respect to the changesin another quantity in an inverse variation

In everyday life, there are situations where an increase in the value of aquantity will cause a corresponding decrease in the value of another quantity

and vice-versa.

E

XAMPLE The volume, V cm

3, of a gas is measured at several pressures, P N/cm2.The table below shows some typical values of V and P.

P (N/cm2) 10 20 30 40

V (cm2) 30 15 10 7.5

State the variation in the value of V if the value of P is(a) doubled (b) halved (c) trebled



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A State the changes in a quantity with respect to the changesin another quantity in an inverse variation

SOLU

TIO

N

P (N/cm2) 10 20 30 40

V (cm2) 30 15 10 7.5


qWhen the value of P is doubled, the value of V is halved.

qWhen the value of P is halved, the value of V is doubled.

qWhen the value of P is trebled, the value of V is multiplied by 1/3 or the value of V is three times less.


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B Determine whether a quantity varies inversely as another quantity


To determine whether y varies inversely as x, just check whether xy is aConstant for all given pairs of (x,y)

Example Determine whether y varies inversely as x, in each of the followingcases.

x 2 4 5 8

y 4 2 1.6 1

x 2 4 8 10

y 16 8 4 3.5

(a)

(b)


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Solution

x 2 4 5 8

y 4 2 1.6 1

(a)

xy 8 8 8 8 A constant

Hence, y varies inversely as x.


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Solution

x 2 4 8 10

y 16 8 4 3.5

(b)

xy 32 32 32 35 Not a constant

Hence, y does not vary inversely as x.


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C Express an inverse variation as an equation in two variables


If y 1 , then xny = k or y = k , where k is the constant of variation.xn xn

EXA

MPLE

(a)Given that y varies inversely as x and y = 24 when x = 2/3, expressy in terms of x.

(b) If P varies inversely as the square root of V and P =

when V = 9, find the equation which relates P and V.


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SO

LUT

ION

(a)Given that y varies inversely as x and y = 24 when x = 2/3, expressy in terms of x.

(a) Given y 1/x.

Then, y = k/x, where k is the constant of variation

Given y = 24 when x = 2/3

= k/( )

2/3

24 x 2/3 = k

k = 16Hence, y = 16

x

24


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CHAPTER 5 : Variations



S

OLUT

ION

(b) If P varies inversely as the square root of V and P = when V = 9, find the equation which relates P and V.

(b) Given P 1V

Then, P = k where k is the constant of variation.V

Given P = 1 when V = 9.

21 = k2 9

1 x 3 = k2

k = 32

Hence, P = 32V

1 = k2 3

5 : Variations


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D Find the value of a variable in an inverse variation given sufficientinformation


5 : Variations

If y 1 , then xny = k or y = k , where k is the constant of variation.xn xn

Hence, given sufficient information, the value of x or y can be determiedby using y = k .xn

EXAMPLE

If y varies inversely as x and y = 8 when x = 2, find(a) the value of y when x = 4,(b) the value of x when y = 10.


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5 : Variations

EXAMPLE

If y varies inversely as x and y = 8 when x = 2, find(a) the value of y when x = 4,(b) the value of x when y = 10.

SOLUTION

Given y 1/x y = k/x


18 = k/2

k = 16Hence, y = 16/x

(a) When x = 4, y = 16/4 = 4

(b) When y = 10, 10 = 16/xx = 16/10x = 1.6


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5 : Variations

ANOTHER WAY

x1y1 = x2y2 (a) Let x1 = 2, y1 = 8 and x2 = 4.2(8) = 4(y2)

y2 = 4

(b) Let x1 = 2, y1 = 8 and y2 = 10.2(8) = x2(10)

x2 = 1.6


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A Represent a joint variation by using the symbol


5 : Variations

A joint variation combines two or more variations involving three or more variables.

(a) y varies directly as p and z means y varies directly as p x z and this iswritten as y pz.

(b) y varies inversely as p and z is y 1 x 1 and this is written as y 1p z pz

(c) y varies directly as p and and inversely as z is y p x 1 and this isz

written as y pz

(d) y varies directly as p2 and and inversely as z3 is y p x 1 and this isz3

written as y p

z3


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B Express a joint variation in the form of an equation


5 : Variations

Example Given y varies directly as x and z2, find the equation relatingx, y and z if y = 6 when x = 2 and z = 1

olution Given y xz2.Then, y = kxz2, where k is the constant of variation

Given x = 2, y = 6 and z =

1.6 = k(2)(1)26 = 2k

k = 3

Hence, the equation relating x, y and z is y = 3xz2.


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C Find the value of a variable in a joint variation givensufficient information


5 : Variations

Example Given A varies inversely as B and the square root of C. If A = 3when B = 2 and C = 4, find the value of C when A = B = 2.

Given A 1.BC

Then, y = k , where k is the constant of variationBC

Given A = 3, B = 2 and C = 4.3 = k 24

k = 12Thus, A = 12

BC

When A = B = 2,2 = 122C

4C= 12

C = 12 = 3

4

C = 32

C = 9

5.0 Variations

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