To solve the problem of determining which expression correctly describes the population of a species that doubles every nine years under ideal conditions, we'll start by understanding the mathematical model of exponential growth.
### Exponential Growth Model:
The population [tex]\( P(t) \)[/tex] at time [tex]\( t \)[/tex] can be expressed using the exponential growth model, particularly when a population doubles over regular intervals. The general form for the population at any time [tex]\( t \)[/tex] is:
[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 elapsed.
- [tex]\( d \)[/tex] is the doubling time.
### Step-by-Step Solution:
1. Identify Given Information:
- Initial population [tex]\( P_0 = 100 \)[/tex].
- Doubling time [tex]\( d = 9 \)[/tex] years.
2. Apply the Exponential Growth Formula:
- Substitute [tex]\( P_0 \)[/tex] and [tex]\( d \)[/tex] into the formula:
[tex]\[
P(t) = 100 \times 2^{\frac{t}{9}}
\][/tex]
3. Evaluation of Provided Expressions:
- [tex]\( 2 \times 100^{9t} \)[/tex]:
This expression does not follow the form of our exponential growth model as it does not use the correct initial population and the doubling mechanism.
- [tex]\( 2 \times 100^{\frac{t}{9}} \)[/tex]:
While this structure uses the exponent [tex]\(\frac{t}{9}\)[/tex], it wrongly incorporates the initial population base [tex]\( 100 \)[/tex] and misplaces the exponent mechanics.
- [tex]\( 100 \times 2^{9t} \)[/tex]:
This expression incorrectly multiplies the exponent by 9 rather than dividing it by the doubling period.
- [tex]\( 100 \times 2^{\frac{t}{9}} \)[/tex]:
This correctly matches our derived formula for exponential growth, using the initial population [tex]\( 100 \)[/tex], the correct base [tex]\( 2 \)[/tex], and the appropriate division by the doubling period [tex]\( 9 \)[/tex].
### Conclusion:
The correct expression for the population of the species [tex]\( t \)[/tex] years after the start, given the initial population of 100 and the doubling time of every nine years, is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{100 \times 2^{\frac{t}{9}}} \][/tex]
