First, let's carefully consider the problem statement. We are given that under ideal conditions, the population of a certain species doubles every nine years. The initial population is 100 individuals. We are tasked with finding an expression that gives the population of the species [tex]\( t \)[/tex] years after the start.
### Step-by-Step Solution:
1. Initial Population ( [tex]\( P_0 \)[/tex] ):
- The initial population at [tex]\( t = 0 \)[/tex] is given as 100 individuals.
2. Doubling Time:
- The population doubles every 9 years. This means that if the initial population is [tex]\( P_0 \)[/tex], then after 9 years, the population will be [tex]\( 2 \times P_0 \)[/tex].
3. Exponential Growth Formula:
- The population growth under ideal conditions can be modeled using the exponential growth formula:
[tex]\[
P(t) = P_0 \times 2^{\frac{t}{T}}
\][/tex]
where [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex],
[tex]\( P_0 \)[/tex] is the initial population,
[tex]\( T \)[/tex] is the doubling time.
4. Substitute Given Values:
- Here, [tex]\( P_0 = 100 \)[/tex] individuals, and the doubling time [tex]\( T = 9 \)[/tex] years.
- Substitute these values into the exponential growth formula:
[tex]\[
P(t) = 100 \times 2^{\frac{t}{9}}
\][/tex]
So the correct expression to calculate the population after [tex]\( t \)[/tex] years is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
Given the provided options, the expression that matches our derived formula is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
Hence, the correct answer is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
