Chapter 21 I Variations ENHANCE

7/23/2019 Chapter 21 I Variations ENHANCE

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CHAPTER 5: VARIATIONS

IMPORTANT CONCEPTS

Direct Variations

Statement Using symbol , Equation wit kas !onstant o"#a$iation

y#a$ies %i$e!tly asx y x y & kx

P #a$ies %i$e!tly as '( P '( P&k'(

M is %i$e!tly )$o)o$tional to

N

M N M& k N

Inverse Variations

Statement Using symbol , Equation wit kas !onstant o"

#a$iationy#a$ies in#e$sely asx

y x

*y &x

k

P #a$ies in#e$sely as '(

P (*

QP& (Q

k

M is in#e$sely )$o)o$tional to

NM N

*M&N

k

Joint Variations

Statement Using symbol , Equation wit kas !onstant o"

#a$iationw#a$ies %i$e!tly asx an%y w xy w & +xy

P #a$ies %i$e!tly as q an% r( P q r( P& +q r(

s #a$ies %i$e!tly t an%

in#e$sely as u s u

ts &u

kt

d #a$ies %i$e!tly as e( an%

in#e$sely as f d f

e(

d &f

ke(

R #a$ies in#e$sely as M an%

NR

NM

*R&

NM

k

4 steps to solve probles involvin! variations:

STEP " : Cange te statement using te symbol

STEP # :-$ite %own te equation !onne!ting te #a$iables using k as te !onstant o"

#a$iation

STEP $ :.in% te #alue o" k

STEP 4 :.in% te #alue o" #a$iable $equi$e%

A% DIRECT VARIATION

Variations 57


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E/AMP0E *1

2i#en tat P#a$ies %i$e!tly as Qan%P&3

*wen Q& (

E4)$essPin te$ms o" Qan% "in% te #alue o" Pwen Q& *5

STEP * 1 Symbol 6 P Q

STEP ( 1 Equation 6 P& +Q , + & !onstant

STEP 3 1 .in% te #alue o" +,3

*& +7(8

3

*& (+

+ &9

*

T:EN, substitute + &9

*in te equationP& +Q

:ENCE, P&

9

*Q

STEP ;1.in% te #alue o"Pwen Q& *5, P&9

* *5

P& 3

E4e$!ises1

* 2i#en taty#a$ies %i$e!tly asxan%y& *< wenx&

!8 E4)$essyin te$ms o"x

%8 .in% te #alue o"ywenx& (


3/21

; It is gi#en tat m #a$ies %i$e!tly as n 3 an% m& 3( wen n& ( E4)$ess min te$ms o"

nan% "in% te #alue o" m wen n&(

*

E/AMP0E ( 1

P#a$ies %i$e!tly asx ( we$ex& < ?y 2i#en tatP& @( weny& *

a8 E4)$essPin te$ms o"x

b8 .in% te #alues o" y wenP& (==

STEP * 1 Symbol 6 P x (

STEP ( 1 Equation 6 P& +x ( , + & !onstant

STEP 3 1 .in% te #alue o" +, substitute an%x& < ? y in te abo#e equation1

P& +7< ?y8 (

Ten substitute P&@( an% y & *, @( & +7< ? *8 (

@( & 39+

+ &9

@(

+ & (

T:EN, substitute + & ( in te equationP& +x (

:ENCE, P& (x (

STEP ;1.in% te #alue o" y wenP& (== by substitutex& < ?y

(== & (7< ?y8 (

*== & 7< ?y8 (

*== & < ?y

*= & < ?y

y& *= < an% y& 6*=

Variations 65


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E/AMP0E ;1

Te table sows some #alues o" te #a$iables d, ean%f

d *( *=

e > (, an%f& ;

*( &;

>k

;5 & 3+

+ & *9

T:EN, substitute + & *9 in te equation d&f

ek

:ENCE, d&f

e*9

STEP ;1.in% te #alue o" m7"8 wen d& *= an% e & (


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( Te table sows some #alues o" te #a$iables w, 4 an% y, su! tat w #a$ies %i$e!tly as te

squa$e $oot o" 4 an% in#e$sely as y

w 4 y

*5 > (

3 D 39

a8 "in% te equation !onne!ting w 4 an% y

b8 !al!ulate te #alue o" D

Ob-ective ./estions

* Te table sows $elation between te #a$iables, 4 an% y I" y 4 , "in% te #alue o" m

A7

!

5

!'C 5

$

%5

(

m ( /

n > ( C 5* *;;

*9

S ( 3

P 3 *

M ; /

Te table sows te $elation between te #a$iables, S, P an% M I" S

MP

!, !al!ulate

te #alue o" 4

A ; 5 C *9 9;

*@ 2i#en tat ) y

x$

an% ) & 9 wen 4 & ( an% y & 3, e4)$ess ) in te$ms o" 4 an% y

A ) &y$

x9 $

) &y9

x$ $

C ) &y9

x$

) &y

x9 $

*5 2i#en tat y #a$ies %i$e!tly as 4 $ an% tat y & 5= wen 4 & ;, e4)$ess y in te$ms o" 4

A y & 4 $ y & m

Te table sows te $elation between te t$ee #a$iables, w, 4 an% y I" w y

x

, !al!ulate

te #alue o" m

A ( ; C 5 *9

(= I" M #a$ies %i$e!tly as te squa$e $oot o" N, te $elation between M an% N is

Variations 7


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A M N M N $ C M N $!

M $

!

N

!

PAST 0EAR .1ESTIONS

"2 SP( #33$*Nov%

2i#en tat# is %i$e!tly )$o)o$tional to n $ an%#& 39, e4)$ess#in te$ms o" n

A#& n $ #& ;n ( C#& >n ( #& *(n (

#2 SP( #33$*Nov%

w ( 3x 5 *5

y ; n

Te table sows some #alues o" te #a$iables, w,xan%ywi! satis"y te $elationsi) w

y

x, !al!ulate te #alue o" n

A 9 > C *( 39

$2 SP( #334*Nov%P#a$ies %i$e!tly as te squa$e $oot o" Q Te $elation betweenPan% Qis

AP (*

Q P (Q CP

(

*

*

QP (

*

Q

42

SP( #334*Nov%

P M r

3 5 ;

9 w >

Te table sows te $elation between te t$ee #a$iables#, man% r 2i#en tat #

r

m, !al!ulate te #alue o" w

A *9 (; C 39 5*

52 SP( #334*J/n%

Variations 7!


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Te $elation between#, nan% ris#r

n3 It is gi#en tat#& ; wen n& 5 an% r& 9

Cal!ulate te #alue o"#wen n& 9; an% r& 3

A *9 (; C 3( ;5

2 SP( #334*J/n%

It is gi#en tat##a$ies in#e$sely wit wan%#& 9 wen w& ( E4)$ess#in te$ms o" w

A # &w

3

w

*(C #& 3w #& *(w

2 SP( #335*Nov%

Te table sows some #alues o" te #a$iablesxan%ysu! taty#a$ies in#e$sely as te

squa$e $oot o"x

x ; *9y 9 3

.in% te $elation betweenyan%x

A y & 3 x x

*(C

(

5

3x

(

>9

x

62 SP( #335*Nov%

It is gi#en tat y #a$ies %i$e!tly as te squa$e $oot o"xan%y& *< wenx& > Cal!ulate te

#alue o"xweny& 3=

A < *5 C (< 3972 SP( #335*Nov%

Te table sows some #alues o" te #a$iables w,xan%ysu! tat w#a$ies %i$e!tly as te

squa$e o" xan% in#e$sely asy

W x y

;= ; (

M 9 ;

Cal!ulate te #alue o" m

A >= ;< C 3= *

Chapter 21 I Variations ENHANCE

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