Cambridge International Mathematics

Example 4 Self Tutor

Consider y=

x^4

5

. State which two variables are directly proportional and
determine the proportionality constantk.

Since y=

x^4

5

, y=^15 x^4 ) y/x^4 and k=^15.

yis directly proportional to the fourth power ofx, and the proportionality

constant k=^15.

Example 5 Self Tutor

SupposeTis directly proportional tod^2 and T= 100 when d=2. Find:

a Twhen d=3 b dwhen T= 200 if d> 0.

Method 1:

T/d^2

) T=kd^2 for some constantk.

When d=2, T= 100

) 100 =k£ 22

) 100 = 4k

) k=25

So, T=25d^2.

a When d=3, T=25£ 32

=25£ 9

= 225

b When T= 200,

200 = 25£d^2

) 8=d^2

) d=2

p

2 fas d> 0 g

Method 2:

T/d^2

a

d 2 3

T 100

dis multiplied by^32

) d^2 is multiplied by

¡ 3

2

¢ 2

) Tis multiplied by

¡ 3

2

¢ 2

) T= 100£

¡ 3

2

¢ 2

= 225

b d 2

T 100 200

Tis multiplied by 2

) d^2 is multiplied by 2

) dis multiplied by

p

2 fd> 0 g

) d=2£

p

2=2

p

2

Example 6 Self Tutor

Theperiodor time for one complete swing of a pendulum is directly proportional to

the square root of its length. When the length is 25 cm, the period is 1 : 00 seconds.

a If the length is 70 cm, find the period to 2 decimal places.

b What would the length be for a period of 2 seconds?

Since is directly

proportional to ,

whatever we multiply

by we must do the

same to.

T

d

T

d

2

2

£ 2

£Ew_

Variation and power modelling (Chapter 30) 609

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Y:\HAESE\IGCSE01\IG01_30\609IGCSE01_30.CDR Monday, 27 October 2008 2:57:34 PM PETER