To determine the correct expression for the population of a species that doubles every nine years under ideal conditions, starting with an initial population of 100 individuals, we need to understand the concept of exponential growth. Let's break down the steps:
1. Identify the Doubling Time and Initial Population:
- The doubling time ([tex]\(T\)[/tex]) is 9 years.
- The initial population ([tex]\(P_0\)[/tex]) is 100 individuals.
2. Formula for Exponential Growth:
- The general formula for population growth under ideal exponential conditions is:
[tex]\[
P(t) = P_0 \times 2^{\frac{t}{T}}
\][/tex]
Here:
- [tex]\(P(t)\)[/tex] is the population at time [tex]\(t\)[/tex] years.
- [tex]\(P_0\)[/tex] is the initial population.
- [tex]\(t\)[/tex] is the time in years.
- [tex]\(T\)[/tex] is the doubling time.
3. Substitute the Given Values:
- Initial population [tex]\(P_0 = 100\)[/tex].
- Doubling time [tex]\(T = 9\)[/tex] years.
- Thus, the formula becomes:
[tex]\[
P(t) = 100 \times 2^{\frac{t}{9}}
\][/tex]
4. Evaluate the Provided Options:
- [tex]\(2 \times 100^{9t}\)[/tex]:
This expression doesn't fit the general exponential growth formula. It incorrectly places the exponential growth factor on the initial population and multiplies by 2.
- [tex]\(2 \times 100^{\frac{t}{9}}\)[/tex]:
This expression incorrectly applies the base-100 exponential growth factor.
- [tex]\(100 \times 2^{9t}\)[/tex]:
This expression incorrectly accelerates the growth rate exponentially every 1/9th of a year.
- [tex]\(100 \times 2^{\frac{t}{9}}\)[/tex]:
This expression matches the correct formulation.
Hence, the correct expression for the given population growth scenario is:
[tex]\[
100 \times 2^{\frac{t}{9}}
\][/tex]
Using this expression, the population at any given time [tex]\(t\)[/tex] years can be accurately calculated.
