Certainly! Let's solve the problem step-by-step.
### Part (a)
We need to find out the population of the state in the year 2000. The given formula for population is:
[tex]\[ A = 21.2 \cdot e^{0.0407t} \][/tex]
Here, [tex]\( A \)[/tex] is the population in millions, and [tex]\( t \)[/tex] is the number of years after 2000. For the year 2000, [tex]\( t = 0 \)[/tex].
Substitute [tex]\( t = 0 \)[/tex] into the formula:
[tex]\[ A = 21.2 \cdot e^{0.0407 \cdot 0} \][/tex]
[tex]\[ A = 21.2 \cdot e^0 \][/tex]
[tex]\[ A = 21.2 \cdot 1 \][/tex]
[tex]\[ A = 21.2 \][/tex]
So, the population of the state in 2000 was [tex]\( 21.2 \)[/tex] million.
### Part (b)
We need to find out when the population of the state will reach 29.8 million. That means we need to find the value of [tex]\( t \)[/tex] when [tex]\( A = 29.8 \)[/tex].
Starting with the equation:
[tex]\[ 29.8 = 21.2 \cdot e^{0.0407t} \][/tex]
Solve for [tex]\( t \)[/tex] by isolating the exponential term:
[tex]\[ \frac{29.8}{21.2} = e^{0.0407t} \][/tex]
Take the natural logarithm (ln) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln\left(\frac{29.8}{21.2}\right) = 0.0407t \][/tex]
[tex]\[ t = \frac{\ln\left(\frac{29.8}{21.2}\right)}{0.0407} \][/tex]
Calculating the value gives approximately [tex]\( t = 8 \)[/tex].
Since [tex]\( t \)[/tex] represents the number of years after 2000, the year when the population will reach 29.8 million is:
[tex]\[ 2000 + 8 = 2008 \][/tex]
### Final Answers
a. In 2000, the population of the state was [tex]\( \boxed{21.2} \)[/tex] million.
b. The population of the state will reach 29.8 million in the year [tex]\( \boxed{2008} \)[/tex].
