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• Notes

Use the fact that the world population was 2560 million in 1950 and 3040 million in 1960 to model the population of the world in the second half of the 20 $\mathrm{th}$ century. (Assume that the growth rate is proportional to the population size).

What is the relative growth rate? Use the model to estimate the world population in 1993 and to predict the population in the year 2020 .

$t=0 \quad \longrightarrow 1950$

$p(t)$ (population)

$p(0)=2560 \quad 1950$

$p(10)=3040 \quad 1960$

$\frac{d p}{d t}=K p$

$K=\frac{1}{10} \ln \frac{3040}{2560} \simeq 0.017185$

The relative growth rate is about

$0.017=1.7 \%$

and the madel is $p(t)=$

$2560 e^{0.017185t}$

$1993 \rightarrow t=43 \rightarrow p(t)=$

$p(43) = 2560 e^{0.017185(43)}$

$p(43)=5360 \mathrm{million}$

$2020 \rightarrow t=70 \rightarrow p(t)=$

$p(70) = 2560 e^{0.017185(70)}$

$p(70)=8524 \mathrm{millibn}$