Question

The table gives estimates of the world population, in millions, from 1750 to 2000. year population year population 1750 790 1900 1650 1800 980 1950 2560 1850 1260 2000 6080 (a) use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. compare with the actual figures

Answer 1

The poopulation exponential model is given by

$P(t)=P_0e^{kt}$

Where, P(t) is the population after year t; Po is the initial population, t is the number of years from the starting year; k is the groth constant.

Given that the population in 1750 is 790 and the population in 1800 is 970, we obtain the population exponential equation as follows:

$970=790e^{50k} \\ \\ \Rightarrow e^{50k}=1.228 \\ \\ \Rightarrow 50k=\ln{1.228}=0.2053 \\ \\ \Rightarrow k=0.0041$

Thus, the exponential equation using the 1750 and the 1800 population values is $P(t)=790e^{0.0041t}$

The population of 1900 using the 1750 and the 1800 population values is given by

$P(t)=790e^{0.041\times150} \\ \\ =790e^{0.6158}=790(1.8511) \\ \\ =1,462$

The population of 1950 using the 1750 and the 1800 population values is given by

$P(t)=790e^{0.041\times200} \\ \\ =790e^{0.821}=790(2.2729) \\ \\ =1,796$

From the table, it can be seen that the actual figure is greater than the exponential model values.