Application: A population model Assume that the growth of the world’s population

Question

Application: A population model

Assume that the growth of the world’s population could be described by the ODE

where

The general solution of this ODE has the form

for some constants K, t0. Throughout the past, we have recorded the following historical data:

a) Consider the new variables y = 106/p, and x = t - 1830. Rewrite the equation in the form y(x) = a0 + a1x.

b) Use the method of least squares to determine a0 and a1.

c) Going back to the original variables, find K and t0. Now you can write out the function p(t) that describes the population p as a function of time t.

d) Use the model to predict the population for t = 1980, t = 2000, and t = 2010. Are you happy with the result? Comparing to the actual population in 2011 which is about 7.1 billion, what would you say about this model?

dt

Explanation / Answer

a0 = 0.102551772 ,a1 = -0.000515053

y = 0.102551772 -0.000515053 *x

c) 106/p = 0.102551772 - 0.000515053 *(t-1830)

d)

t = 1980

106/p = 0.102551772 - 0.000515053 *(1980-1830)

p = 4190.75

t = 2000

106/p = 0.102551772 - 0.000515053 *(2000-1830)

p = 7070.08

t = 2010

106/p = 0.102551772 - 0.000515053 *(2010-1830)

p = 10769.9

t = 2011

p=11364.6 million

p t y x 545 1650 0.194495 -180 623 1700 0.170144 -130 728 1750 0.145604 -80 906 1800 0.116998 -30 1171 1850 0.090521 20 1608 1900 0.06592 70 1834 1920 0.057797 90 2295 1940 0.046187 110 3003 1960 0.035298 130

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