# The population of a region is growing exponentially. There were 10 million people in 1980 (when t=0) and 75 million people in 1990. Find an exponential model for the population (in millions of people) at any time tt, in years after 1980. P(t)=? What population do you predict for the year 2000? Predicted population in the year 2000 =million people. What is the doubling time?

Question
Exponential models
The population of a region is growing exponentially. There were 10 million people in 1980 (when t=0) and 75 million people in 1990. Find an exponential model for the population (in millions of people) at any time tt, in years after 1980.
P(t)=?
What population do you predict for the year 2000?
Predicted population in the year 2000 =million people. What is the doubling time?

2021-02-21
Given:
p(0)=10 million ... at t=0, 1980
p(10)=75 million ... at t=10 1990
So, p(0)=p_0=10 million
$$\displaystyle\Rightarrow{p}{\left({t}\right)}={p}_{{0}}{e}^{{{k}{t}}}$$
$$\displaystyle\Rightarrow{p}{\left({t}\right)}={10}{e}^{{{k}{t}}}$$
when t=10, (1990) $$\displaystyle\Rightarrow{p}{\left({10}\right)}={75}$$
$$\displaystyle\Rightarrow{p}{\left({10}\right)}={10}{e}^{{{10}{k}}}\Rightarrow{75}={10}{e}^{{{10}{k}}}$$
$$\displaystyle\Rightarrow{e}^{{{10}{k}}}={\frac{{{75}}}{{{10}}}}$$
$$\displaystyle\Rightarrow{e}^{{{10}{k}}}={7.5}$$
Take log on both sides
$$\displaystyle\Rightarrow{10}{k}={{\log}_{{e}}{\left({7.5}\right)}}$$
$$\displaystyle{10}{k}={2.02}$$
$$\displaystyle{k}={\frac{{{2.02}}}{{{10}}}}$$
$$\displaystyle{k}={0.202}$$
$$\displaystyle\Rightarrow{k}={0.2}$$
$$\displaystyle{p}{\left({t}\right)}={10}{e}^{{{0}\cdot{2}{t}}}$$
predicted population in year 2000
$$\displaystyle\Rightarrow{t}={20}$$
$$\displaystyle\Rightarrow{p}{\left({20}\right)}={10}{e}^{{{0}\cdot{2}{\left({20}\right)}}}$$
$$\displaystyle\Rightarrow{p}{\left({20}\right)}={10}{e}^{{4}}$$
$$\displaystyle\Rightarrow{p}{\left({20}\right)}={545.98}\approx{546}$$
Population in 2000 p(20)= 546 million
we, $$\displaystyle{p}{\left({t}\right)}={10}{e}^{{{0}\cdot{2}{t}}}$$
Now,
The population during doubling time $$\displaystyle{p}{\left({t}\right)}={2}\times{10}$$ at t=0
$$\displaystyle{p}{\left({t}\right)}={20}$$
$$\displaystyle\Rightarrow{20}={100}{e}^{{{0}\cdot{2}{t}}}$$
$$\displaystyle\Rightarrow{e}^{{{0}\cdot{2}{t}}}={2}$$
$$\displaystyle{0}\cdot{2}{t}={\ln{{\left({2}\right)}}}$$
$$\displaystyle{0}\cdot{2}{t}={0.69}$$
$$\displaystyle{t}={\frac{{{0.69}}}{{{0.2}}}}$$
$$\displaystyle{t}={3.47}$$ doubling time

### Relevant Questions

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