Use the fact that the world population was 2560 million in 1950 and 3040 million

Question

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 20th 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 1990 and to predict the population in the year 2030. SOLUTION We measure the time t in years and let t = 0 in the year 1950. We measure the population P(t) in millions of people. Then P(0) = 2560 and P(10) = 3040. Since we are assuming that dP/dt = kP, this theorem gives the following. (Round to six decimal places.) P(t) = P(0)ekt = ekt P(10) = e10k = 3040 k = 1/10 ln(3040/2560) The relative growth rate is about 1.7% per year and the model is P(t) = We estimate that the world population in 1990 was P( ) = million (rounded to the nearest million). The model predicts that the population in 2030 will be P( ) = million (rounded to the nearest million). The graph shows that the model is fairly accurate to date (the dots represent the actual population), so the estimate for 1990 is quite reliable. for 2030 is riskier.

Explanation / Answer

world population in 1990

t=1990-1950=40

p(40)=2560e0.017185*40

p(40)=2560*1.98854

p(40)=5090.66

p(40)=5091 million

in 2030.

t=2030-1950=80

p(80)=2560e0.017185*80

p(80)=2560*3.95428

p(80)=10122.96

p(80)=10123 million

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