Answer
[tex]N(t)=3780000(1.027)^t[/tex]
Explanation
To solve this, we are going to use the exponential growth function:
[tex]f(t)=a(1+b)^t[/tex]
where
[tex]f(t)[/tex] is the final population after [tex]t[/tex] years of growth
[tex]a[/tex] is the initial population
[tex]b[/tex] is the growth rate in decimal form
[tex]t[/tex] is the time in years
In our model [tex]t[/tex] will be the number of years after 2004, so the number of years after 2004 is 0 and the number of years after 2009 is 5. The initial population is, therefore, the number of licensed drivers in 2004, so [tex]a=3.78million=3780000[/tex]. The final population is the number of licensed drivers in 2009, so [tex]f(t)=4.32million=4320000[/tex]. Let's use those facts to calculate the growth rate and complete our model:
For the final population
[tex]t=5[/tex], [tex]a=3780000[/tex], and [tex]f(t)=4320000[/tex]. Let's replace the values in our function
[tex]f(t)=a(1+b)^t[/tex]
[tex]4320000=3780000(1+b)^5[/tex]
[tex]\frac{4320000}{3780000} =(1+b)^5[/tex]
[tex]1+b=\sqrt[5]{\frac{4320000}{3780000}}[/tex]
[tex]b=\sqrt[5]{\frac{4320000}{3780000}}-1[/tex]
[tex]b=0.027[/tex]
Now we can complete our model
[tex]f(t)=a(1+b)^t[/tex]
[tex]N(t)=3780000(1+0.027)^t[/tex]
[tex]N(t)=3780000(1.027)^t[/tex]
where [tex]t[/tex] is the number of years after 2004
