# In 1992, the city of Chicago had 6.5 million people. In 2000 they project Chicago will have 6.6 million people. If Chicago's population grows exponentially, how many people will live in Chicago in 2005?

Jun 24, 2015

Chicago's population in 2005 will be approximately 6.7 million people.

#### Explanation:

If the population grows exponentially, then its formula has the following form:

$P \left(t\right) = A \cdot {g}^{t}$ with $A$ the initial value of the population, $g$ the growth rate and $t$ the time passed since the beginning of the problem.

We start the problem in 1992 with a population of $6.5 \cdot {10}^{6}$ and in 2000 -8 years later- we expect a population of $6.6 \cdot {10}^{6}$.
Therefore, we have
$A = 6.5 \cdot {10}^{6}$
$t = 8$
If we consider one million people as the unit of the problem, we have
$P \left(8\right) = 6.5 \cdot {g}^{8} = 6.6$

$\rightarrow {g}^{8} = \frac{6.6}{6.5} \rightarrow g = \sqrt[8]{\frac{6.6}{6.5}}$

We are looking for the population in 2005, 13 years after the beginning of the problem:

$P \left(13\right) = 6.5 \cdot {g}^{13} = 6.5 \cdot {\left(\frac{6.6}{6.5}\right)}^{\frac{13}{8}} = 6.6 \cdot {\left(\frac{6.6}{6.5}\right)}^{\frac{5}{8}} = 6.663280 \approx 6.7$