To determine the population of a species that doubles every nine years, starting with 100 individuals, we can use an exponential growth model. Let's break this down step by step:
1. Initial Population and Doubling Time:
- Initial population, [tex]\( P_0 = 100 \)[/tex] individuals.
- Doubling time, [tex]\( T_d = 9 \)[/tex] years.
2. General Form of Exponential Growth:
The population at time [tex]\( t \)[/tex] can be expressed using the general exponential growth formula:
[tex]\[
P(t) = P_0 \cdot 2^{\frac{t}{T_d}}
\][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( T_d \)[/tex] is the doubling time,
- [tex]\( t \)[/tex] is the time elapsed.
3. Substitute the Given Values:
Here, [tex]\( P_0 = 100 \)[/tex] and [tex]\( T_d = 9 \)[/tex]. Substituting these values into the formula, we get:
[tex]\[
P(t) = 100 \cdot 2^{\frac{t}{9}}
\][/tex]
Thus, the correct mathematical expression that gives the population of the species [tex]\( t \)[/tex] years after the start is:
[tex]\[
100 \times 2^{\frac{t}{9}}
\][/tex]
Conclusion:
Among the given options, the correct one is:
[tex]\[
100 \times 2^{\frac{t}{9}}
\][/tex]
