A Concrete Approach to Mathematical Modelling by Mike Mesterton-Gibbons

By Mike Mesterton-Gibbons

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S. population in millions, 1790-1850. 3 Data for x(i) are taken from Historical Statistics of the United States, Colonial Times to 1970, Bicenten› nial Edition, Part I , p. S. , 1975). S. 2 for x(2). Can the same model predict population magnitude in later years? 57) are compared wit h observed values for the years 1 8 5 0 - 1 9 7 0. 57) soon fail s to describe population growth wit h acceptable accuracy. 24 for the years 1 8 6 0 - 1 8 8 0, suggesting the improved model \ 3 1 i ? 24(/-7) ( ( > 6 5 But although thi s is correct to tw o significant figures for the years 1 8 6 01880, and even to one significant figure for 1850 and 1890, it seriously overestimates the magnitude of the population in the present century.

It would be equally meaningless t o measure th e weight of the output i n tonnes, because a factory producing 1000 glass paperweights would have equalled the output of one producing several thousand scientific calculators. I n avoiding such absurdities, the traditiona l method of incorporatin g diverse products int o a single measure of output has been t o count each according t o it s monetary value. Even here there’s a difficulty , however, because it’ s c o m m on knowledge that a dollar won’t buy as much in the year 2000 as it would have bought in 1950; or, more gener› ally, that it won’t buy as much today as it would have bought / years ago.

Notice that, for given tractiv e force and drag, large Μ is associated wit h small acceleration, and small Μ is associated wit h large acceleration. Μ is therefore a measure of the boat’s reluctance t o change speed, or inertia. We call Μ the mass. Before proceeding, we pause t o record a detail fro m physics. Heavy objects have greater reluctance t o change speed than light ones (tr y push› ing), hence greater mass. But heavier objects, by definition , have greater weight, denoted by W. Indeed the relationship between weight and mass i s a simple one of proportionality , namely, W = Mg.

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