Cambridge International Mathematics

OTHER DIRECT VARIATION

The formula for finding the areaAof a circle of radiusr,is

A=¼r^2.

r 0 1 2 3 4

A 0 3 : 14 12 : 57 28 : 27 50 : 27

If we graphAagainstrwe get the graph alongside. The

graph is not a straight line, but rather is part of aparabola.

However, if we graphAagainstr^2 we get this graph:

r 0 1 2 3

r^20149

A 0 3 : 14 12 : 57 28 : 27

Since the graph ofAagainstr^2 is a straight line through the

origin O,Ais directly proportional tor^2.

We write A/r^2 , and so A= kr^2 wherek is the

proportionality constant. In this case we know k=¼.

Notice from the table that asris doubled from 1 to 2 ,r^2 is

multiplied by 4. So,Ais also increased by a factor of 4.

Example 3 Self Tutor

Consider the table of values: x 2 4 6 8

y 6 24 54 a

a By finding

y

x^2

for each point, establish that y/x^2.

b Write down the rule connectingyandx.

c Find the value ofa.

a When x=2,

y

x^2

=

6

22

=1: 5

When x=4,

y

x^2

=

24

42

=1: 5

When x=6,

y

x^2

=

54

62

=1: 5

y

x^2

=1: 5 in each case, and so y=1: 5 x^2. Hence y/x^2.

b y=1: 5 x^2

c When x=8, y=a ) a=1: 5 £ 82 =96.

r

A

40

20

5

10

30

50

O

A

30

10

20

5 10

r^2

O

608 Variation and power modelling (Chapter 30)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_30\608IGCSE01_30.CDR Monday, 27 October 2008 2:57:32 PM PETER