Cambridge International Mathematics

Example 12 Self Tutor

A small glass ball is rolled down a sloping

sheet of hard board. At timetseconds it has

rolled a distancedcm. The results are:

t 0 1 2 3 4 5

d 0 0 : 2 0 : 8 1 : 8 3 : 2 5 : 0

To show the model is quadratic, we graphdagainstt^2 :

Since this graph is linear, the model is indeed quadratic.

d=kt^2 wherekis the gradient of the line.

k=

5 ¡ 0

25 ¡ 0

=^15 =0: 2

) d=0: 2 t^2

Example 13 Self Tutor

height (hcm) 8 12 16

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

Assuming that m_hn for some integern, find an equation

connectingmandh.

We are given that m_hn,som=khn wherekandnare constants.

When h=8, m=61: 44 ,so 61 :44 =k£ 8 n ...... (1)

When k=16, m= 491: 52 ,so 491 :52 =k£ 16 n ...... (2)

Dividing (2) by (1) gives

k£ 16 n

k£ 8 n

=

491 : 52

61 : 44

)

¡ 16

8

¢n

=8

) 2 n=2^3

) n=3

Using (1), 61 :44 =k£ 83

) k=

61 : 44

512

) k=0: 12

) the equation is m=0: 12 h^3

We check this model using the third data point: 0 : 12 £ 123 = 207: 36 X

5 10152025

1

2

3

4

5

d

tX

Copies of the Eiffel Tower are sold in different sizes all over

Paris. Of those made of brass, measurement of the height and

mass of several samples was made:

Show that the model is quadratic and find its

equation.

Variation and power modelling (Chapter 30) 617

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