To determine the expression that gives the population of a species [tex]$t$[/tex] years after the start, given that the population doubles every nine years and starts with 100 individuals, we need to formulate a mathematical model for population growth.
1. The population doubling every nine years suggests an exponential growth model. The general form of an exponential growth function is:
[tex]\[
P(t) = P_0 \times a^{t/T}
\][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( a \)[/tex] is the growth factor,
- [tex]\( T \)[/tex] is the doubling period,
- [tex]\( t \)[/tex] is the time in years.
2. Since the population doubles every nine years, the doubling period [tex]\( T \)[/tex] is 9 years, and the growth factor [tex]\( a \)[/tex] is 2 (because the population doubles). Thus, our model becomes:
[tex]\[
P(t) = P_0 \times 2^{t/9}
\][/tex]
3. Given that the initial population [tex]\( P_0 \)[/tex] is 100 individuals, we substitute [tex]\( P_0 = 100 \)[/tex] into the model:
[tex]\[
P(t) = 100 \times 2^{t/9}
\][/tex]
Therefore, the correct expression for the population of the species [tex]$t$[/tex] years after the start is:
[tex]\[
100 \times 2^{\frac{t}{9}}
\][/tex]
So, the correct choice is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
