The form of the exponential function is
[tex]y=a(1\pm r)^x[/tex]a is the initial amount
r is the percent of increase or decrease in decimal
We use + when it is growth
We use - when it is decay
The given exponential function is
[tex]P=494.29(1.03)^t[/tex]By comparing it with the model above
a = 494.29 ---- initial population
1 + r = 1.03
a)
Since 1.03 is greater than 1, then
The model represents exponential growth
b)
Since 1 + r = 1.03, then subtract 1 from each side to find r
[tex]\begin{gathered} 1+r=1.03 \\ 1-1+r=1.03-1 \\ r=0.03 \end{gathered}[/tex]Change it to percent by multiplying it by 100%
[tex]0.03\times100\text{ \%=3\%}[/tex]The annual percentage of increase is 3%
c)
If the population is 590,000, then substitute P by this value and solve to find t
[tex]590000=494.29(1.03)^t[/tex]Divide each side by 494.29
[tex]\frac{590000}{494.29}=(1.03)^t[/tex]Insert ln for both sides
[tex]\ln (\frac{590000}{494.29})=\ln (1.03)^t[/tex]Use the rule
[tex]\ln (a)^b=b\ln (a)[/tex][tex]\ln (\frac{590000}{494.29})=t\ln (1.03)[/tex]Divide both sides by ln(1.03) to find t
[tex]\begin{gathered} \frac{\ln (\frac{590000}{494.29})}{\ln (1.03)}=t \\ 239.68344=t \end{gathered}[/tex]Round it to the nearest years
t = 240
The population will be 590,000 after 240 years
