Question

The population of a Midwestern city follows the exponential law. If the population decreased from 900,000 to 800,000 from 2008 to 2010, what will the population be in 2012?Please use number steps to explain!!

Answer 1

Answer:

Concept:

The exponential function is given below as

$\begin{gathered} y=ab^x \\ a=initial\text{ population} \\ x=number\text{ of years} \end{gathered}$

STEP 1:

In the year 2008, the population is

$\begin{gathered} a=900000 \\ x=0 \end{gathered}$

In the year 2009 ,the population is

$\begin{gathered} =800,000 \\ x=2 \end{gathered}$

By substituting the values, we will have

$\begin{gathered} y=ab^x \\ 800000=900000b^2 \\ divide\text{ both sides by 900,000} \\ \frac{800000}{900000}=\frac{900000b^2}{900000} \\ b^2=\frac{8}{9} \\ b=\sqrt{\frac{8}{9}} \\ b=0.9428 \end{gathered}$

Step 2:

To figure out the population in 2012,

The value of x will be

$\begin{gathered} x=4 \\ (2008\text{ to 2012\rparen} \\ =2012-2008=4years \end{gathered}$$\begin{gathered} y=ab^x \\ y=900000(\sqrt{\frac{8}{9}})^4 \\ y=9000000\times\frac{64}{81} \\ y=711,111 \end{gathered}$

Hence,

The population in 2012 will be

$\Rightarrow711,111$