To model the population growth of this species, we can use the exponential growth formula:

$P(t) = P_0 \cdot e^{kt}$

Where:

• $P(t)$ is the population at time $t$.
• $P_0$ is the initial population.
• $k$ is the growth rate constant.
• $t$ is the time in years.
• $e$ is the base of the natural logarithm (approximately 2.718).

Given:

• The initial population $P_0 = 100$.
• The population doubles every 9 years, so after 9 years, the population becomes 200.

Step 1: Determine the growth rate constant $k$

We know that the population doubles every 9 years. Thus, at $t = 9$, the population is twice the initial population, meaning:

$P(9) = 2 \cdot P_0$

Substituting into the exponential growth formula:

$2 \cdot P_0 = P_0 \cdot e^{k \cdot 9}$

Canceling $P_0$ from both sides (since $P_0 \neq 0$):

$2 = e^{9k}$

Taking the natural logarithm ($\ln$) of both sides to solve for $k$:

$\ln(2) = 9k$

$k = \frac{\ln(2)}{9}$

Now, $\ln(2) \approx 0.693$, so:

$k \approx \frac{0.693}{9} \approx 0.077$

Step 2: Write the population formula

Now that we have $k \approx 0.077$, we can substitute it back into the exponential growth formula:

$P(t) = 100 \cdot e^{0.077t}$

So, the expression that models the population $t$ years after the start is:

$P(t) = 100 \cdot e^{0.077t}$

Summary

The expression that gives the population of the species $t$ years after the start, under ideal conditions, is:

$P(t) = 100 \cdot e^{0.077t}$

Would you like further details on how this model works or a step-by-step breakdown of any part of the solution?

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Related questions:

1. How do you calculate the population after 18 years?
2. What would the population be after 27 years?
3. How can we use the model to find the time it takes for the population to triple?
4. How would the model change if the population doubled every 5 years instead of 9 years?
5. What other growth models might apply if the conditions change (e.g., limited resources)?

Tip: When working with exponential growth or decay, it's useful to remember that the rate of change depends on the current population, and doubling or halving times can help identify the growth rate constant.