Question

According to the model, when will the population be 323 million people? Be sure to round your answer to the nearest whole year.

Answer 1

We are given that the population of the U.S is modeled by the following exponential equation:

`P(t)=304(1.011)^(t-2008)`

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:

`323=304(1.011)^(t-2008)`

Now, to solve for "t" we will divide by 304 on both sides:

`(323)/(304)=(1.011)^(t-2008)`

Now we take "ln" in both sides:

`\ln (323)/(304)=ln(1.011)^(t-2008)`

Now we use the following property of logarithms:

`\ln (a^x)=x\ln a`

Applying the property we get:

`\ln (323)/(304)=(t-2008)ln(1.011)`

Now we divide both sides by "ln(1.011)":

`(1)/(\ln\mleft(1.011\mright))\ln (323)/(304)=(t-2008)`

Now we add 2008 to both sides:

`(1)/(\ln(1.011))\ln (323)/(304)+2008=t`

Now we solve the operations:

`2013.5=t`

Approximating to the nearest whole year we get:

`2014=t`

Therefore, in the year 2014, the population will be 323 million.