Question

Under ideal conditions, the population of a certain species doubles every nine years. If the population starts with 100 individuals, which of the following expressions would give the population of the species [tex]t[/tex] years after the start, assuming that the population is living under ideal conditions?

A. [tex]2 \times 100^{9 t}[/tex]
B. [tex]2 \times 100^{\frac{t}{9}}[/tex]
C. [tex]100 \times 2^{9 t}[/tex]
D. [tex]100 \times 2^{\frac{t}{9}}[/tex]



Answer :

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]