ANSWER:
a. 21.2
b. 2008
STEP-BY-STEP EXPLANATION:
We have that the function that models the population is the following:
[tex]A=21.2\cdot e^{0.0407t}[/tex]a.
In the year 2000, t = 0, we substitute and calculate the population, just like this:
[tex]\begin{gathered} A=21.2\cdot e^{0.0407\cdot0} \\ \\ A=21.2\cdot e^0 \\ \\ A=21.2\cdot1 \\ \\ A=21.2 \end{gathered}[/tex]In 2000, the population of the state was 21.2 million.
b.
Now, in this case A = 29.8, we solve for t, just like this:
[tex]\begin{gathered} 29.8=21.2\cdot \:e^{0.0407\cdot \:t} \\ \\ e^{0.0407\cdot\:t}=\frac{29.8}{21.2} \\ \\ 0.0407t=\ln\left(\frac{29.8}{21.2}\right) \\ \\ t=\frac{\ln\left(\frac{29.8}{21.2}\right)}{0.0407} \\ \\ t=8.37 \end{gathered}[/tex]Therefore, it would be 8 years after 2000. The population of the state will reach 29.8 million in the year 2008