Solution:
An exponential function is generally expressed as
[tex]\begin{gathered} y=a(b)^t\text{ ----- equation 1} \\ \end{gathered}[/tex]Given that in a research study, a colony of penguins had a population of 20,000.
This implies that
[tex]\begin{gathered} when\text{ t=0,} \\ 20,000=ab^0 \\ \Rightarrow20000=a\times1\text{ \lparen where b}^0=1) \\ thus, \\ a=20000 \end{gathered}[/tex]Substitute the value of a into equation 1.
Thus,
[tex]y=20000(b)^t\text{ ----- equation 2}[/tex]One year later, it had a population of 21,200. This implies that when t equals 1, we substitute the values of 21200 and 1 for y and t respectively into equation 2.
This gives
[tex]\begin{gathered} 21200=20000(b)^1 \\ \Rightarrow21200=20000b \\ divide\text{ both sides by the coefficient of b, which is b.} \\ thus, \\ \frac{21200}{20000}=\frac{20000b}{20000} \\ \Rightarrow b=1.06 \end{gathered}[/tex]Substitute the obtained value of b into equation 2.
Thus, the expression that best models the population is
[tex]20,000(1.06)^t[/tex]Assuming the colony grows at the same rate, the population of the colony after 4 years is evaluated by solving for y when the value of t is 4.
Thus,
[tex]\begin{gathered} y=20,000(1.06)^t \\ when\text{ t=4, we have} \\ y=20,000(1.06)^4 \\ =20000\times(1.06)^4 \\ =20000\times1.26247696 \\ \Rightarrow y=25249.5392 \\ \therefore y=25250\text{ \lparen nearest whole number\rparen} \end{gathered}[/tex]Hence, after 4 years the population of the colony will be 25250 penguins (nearest whole number).