To solve this problem, we need to understand the population growth under ideal conditions. The population doubles every nine years. Let [tex]\( t \)[/tex] be the number of years after the start.
Initially, the population is 100 individuals.
In exponential growth scenarios where the population doubles at a regular interval, the population [tex]\( P(t) \)[/tex] at time [tex]\( t \)[/tex] can be calculated using the formula:
[tex]\[ P(t) = P_0 \times 2^{\frac{t}{d}} \][/tex]
where:
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( t \)[/tex] is the time in years,
- [tex]\( d \)[/tex] is the doubling period in years.
Here, [tex]\( P_0 \)[/tex] is 100, and [tex]\( d \)[/tex] is 9 years.
Substituting these values into the formula gives us:
[tex]\[ P(t) = 100 \times 2^{\frac{t}{9}} \][/tex]
So, the correct expression for the population after [tex]\( t \)[/tex] years is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
Thus, the correct choice matches the expression we derived:
[tex]\[ \boxed{100 \times 2^{\frac{t}{9}}} \][/tex]
Therefore, the answer is the fourth option.
