To address this question, let's assume the population follows an exponential growth model:

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

where:

• $P(t)$ is the population at time $t$,
• $P_0$ is the initial population,
• $k$ is the growth rate,
• $t$ is time in years.

Given Information:

1. $P(3) = 10,000$ (Population after 3 years).
2. $t = 10$ for predictions.
3. We need to find:
   • $P_0$ (initial population),
   • $P(10)$ (population after 10 years),
   • $P'(10)$ (rate of population growth at $t = 10$).

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Step 1: Solve for $P_0$

From the equation: $P(3) = P_0 e^{3k}$ Substitute $P(3) = 10,000$: $10,000 = P_0 e^{3k}$

Next, we need to solve for $k$, so we require another condition or method to estimate it.

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Do you have any additional information about $k$ or another population value at a different time? This will help solve for both $P_0$ and $k$.