Cambridge International Mathematics

v 10 20 30 40

R 0 : 5 4 13 : 5 32

7 The designer of a car windscreen needs to find the relationship between

the air resistanceRand the velocityvkm/h of the car. He carries out

an experiment and the results are given in the table alongside:

R_vn.

a Plot the graphs ofRagainstv, Ragainstv^2 , and Ragainstv^3.

b Hence deduce the approximate model forRin terms ofv.

These are equations

of the form y=axb.

If b> 0 , we havedirect variation.

If b< 0 , we have

So far we have only considered data which a power model fitsexactly. We now consider data for which the

‘best’ model must be fitted. In such a case we use technology to help find the model.

If we suspect two variables are related by a power model, we should:

² Graphyagainstx. We look for either:

I a direct variation curve which passes

through the origin.

I

xandy-axes are both asymptotes.

² Use thepower regressionfunction on your calculator.

Instructions for this can be found on page 28.

For example, consider the data inExample 13:

8 12 16

mass (mg) 61 : 44 207 : 36 491 : 52

A graph of the data suggests a power model is reasonable.

Using the power regression function on a graphics calculator, we

obtain the screen dump opposite.

So, m=0: 12 h^3.

The coefficient of determination r^2 =1indicates the model is a

perfect fit.

O

y

x O

y

x

m(g)

O h(cm)

()8 ¡61 44,.

()12 ¡207.36,

()16 ¡491.52,

D POWER MODELLING

Variation and power modelling (Chapter 30) 619

From previous experiences, the designer expects that

The direct and inverse variations we have studied are all examples ofpower models.

inverse variation.

an inverse variation curve for which the

height (h )cm

[2.13, 11.2]

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Y:\HAESE\IGCSE01\IG01_30\619IGCSE01_30.CDR Tuesday, 18 November 2008 11:59:10 AM PETER