We are given that the population of the U.S is modeled by the following exponential equation:
[tex]P(t)=304(1.011)^{t-2008}[/tex]
We want to determine the time "t" for which the population is 323 million people. To do that we will replace the value of P(t) for 323, since P is given as millions of people. Replacing we get:
[tex]323=304(1.011)^{t-2008}[/tex]
Now, to solve for "t" we will divide by 304 on both sides:
[tex]\frac{323}{304}=(1.011)^{t-2008}[/tex]
Now we take "ln" in both sides:
[tex]\ln \frac{323}{304}=ln(1.011)^{t-2008}[/tex]
Now we use the following property of logarithms:
[tex]\ln (a^x)=x\ln a[/tex]
Applying the property we get:
[tex]\ln \frac{323}{304}=(t-2008)ln(1.011)[/tex]
Now we divide both sides by "ln(1.011)":
[tex]\frac{1}{\ln\mleft(1.011\mright)}\ln \frac{323}{304}=(t-2008)[/tex]
Now we add 2008 to both sides:
[tex]\frac{1}{\ln(1.011)}\ln \frac{323}{304}+2008=t[/tex]
Now we solve the operations:
[tex]2013.5=t[/tex]
Approximating to the nearest whole year we get:
[tex]2014=t[/tex]
Therefore, in the year 2014, the population will be 323 million.
