Cambridge International Mathematics

Mass (mkg) 0 : 02 0 : 5 4 20 50 100 500

Heat required (hJoules) 25 170 630 1670 2500 4200 16 750

a Obtain a power model for this data.

b Is the power model appropriate? Explain your answer.

c What would be the heat required by an elephant of mass 5 tonnes?

4 Johann Kepler is a very famous man in the history of astronomy and mathematics. He used data from

observations of planetary orbits to show that these motions are not random but rather obey certain

mathematical laws which can be written in algebraic form. He took the Earth as his “base unit”, so

the orbital periods are given as multiples of one Earth year, and orbital radii as multiples of one Earth

orbit. Some of his observational data are given in this table.

Planet Mercury Venus Earth Mars Jupiter Saturn

Orbital period (T) 0 : 241 0 : 615 1 : 000 1 : 881 11 : 862 29 : 457

Orbital radius (R) 0 : 387 0 : 723 1 : 000 1 : 542 5 : 202 9 : 539

a Obtain a power model for this data.

b Explain whether the power model is appropriate.

c

5

P 1015 2000 3000 4000 5000 6000

V 1000 615 460 375 320 281

a Find the power model of best fit.

b Estimate the volume when: i P= 2400hPa ii P= 7000hPa

Discovery The pendulum

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Apendulumcan be made by tying an object such as a mobile phone

to a shoe lace and allowing the object to swing back and forth when

supported by the string.

We will find theperiod, which is the time taken for one complete

oscillation, back and forth, for various lengths of the pendulum.

The maximum angleμof the pendulum should be 15 o. The length of

the pendulum is the distance from the point of support to the object’s

centre of mass.

lcm

point of

support

q°

object

If you compress a gas quickly, it gets hot. The more you

compress it, the hotter it gets. Just think how hot a bike

pump gets when you pump up a tyre quickly. In such cases,

Boyle’s law which connects volume ( ml) and pressure

( hectopascals) no longer applies. To find a new model for

this case, data has been obtained. It is:

V

P

The object of this discovery is to determine any rule which may

connect the period of the pendulum ( secs) with its length ( cm).Tl

Variation and power modelling (Chapter 30) 621

What is the resulting equation when the equation ofais cubed?

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Y:\HAESE\IGCSE01\IG01_30\621IGCSE01_30.CDR Wednesday, 29 October 2008 4:03:45 PM PETER