Given
The population of the city follows an exponential law.
Population at 2008 = 900000
Population at 2010 = 800000
The population decrease follows an exponential decay law which is defined as:
[tex]\begin{gathered} N(t)\text{ = N}_0e^{kt} \\ Where\text{ k is the decay rate} \\ N_0\text{ is the initial amount} \end{gathered}[/tex]
For the given problem:
[tex]N_o\text{ = 900000}[/tex]
After 2 years (t =2), the population decreased to 800000. Hence we can write:
[tex]\begin{gathered} 800000\text{ = 900000e}^{2k} \\ e^{2k}\text{ = }\frac{800000}{900000} \\ e^{2k}\text{ = }\frac{8}{9} \\ 2k\text{ = }\ln(\frac{8}{9}) \\ k\text{ = -0.0589} \end{gathered}[/tex]
Hence, the equation the represents the population (N(t)) as a function of year (t):
[tex]N(t)\text{ = 900000e}^{-0.0589t}[/tex]
The population in 2012 is the population after 4 years ( t =4)
Substituting into the formula and solving:
[tex]\begin{gathered} N(t=4)\text{ =900000e}^{-0.0589\times4} \\ =\text{ 711086.9845} \\ \approx\text{ 711087} \end{gathered}[/tex]
Hence, the population in 2012 would be 711087
