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Solving Word Problems Involving Applications of Exponential Functions to Growth and Decay

The population of a certain county can be modeled by the equation:

$P(t)=50e^{0.02t}$

Where $P$ is the population in millions, $t$ is the number of years since 1900. Find when the population is 100 million, 200 million, and 400 million. What do you notice about these time periods?

100 million

$\frac{100,000,000}{50}=\frac{50e^{0.02t}}{50}$

$2,000,000=e^{0.02t}$

$ln(2,000,000)=ln(e^{0.02t})$

$\frac{14.51}{0.02}=\frac{0.02t}{0.02}$

$725.43=t$

200 million

$200,000,000=50e^{0.02t}$

$4,000,000=e^{0.02t}$

$ln(4,000,000)=ln(e^{0.02t})$

$15.20=0.02t$

$760.09=t$

400 million

$ln(8,000,000)=ln(e^{0.02t})$

$t=794.75$

$794.75-760.09=34.66\ yrs$

$760.09-725.43=34.66\ yrs$

Population doubles every 34.66 yrs