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The problem presents a fish population \( P(t) \) approximated by the equation:

\[
P(t) = 15e^{0.6t}
\]

where \( t \) is in months. You're asked to:

1. Calculate \( P(12) \).
2. Calculate \( P'(12) \) (the derivative of the population function evaluated at \( t = 12 \)).
3. Explain what these values mean with respect to the population, including units.

Let's break this down:

### 1. Finding \( P(12) \):
We plug \( t = 12 \) into the function \( P(t) \):

\[
P(12) = 15e^{0.6(12)}
\]

### 2. Finding \( P'(t) \):
The derivative of \( P(t) = 15e^{0.6t} \) with respect to \( t \) is:

\[
P'(t) = 15 \times 0.6 \times e^{0.6t} = 9e^{0.6t}
\]

Now we can evaluate \( P'(12) \):

\[
P'(12) = 9e^{0.6(12)}
\]

### Let's compute both values.The results are as follows:

1. \( P(12) \approx 20,091 \): This represents the population of fish after 12 months. The units here are "number of fish."
   
2. \( P'(12) \approx 12,055 \): This represents the rate of change of the fish population at 12 months, indicating that the population is increasing by approximately 12,055 fish per month at \( t = 12 \).

Would you like more details on these calculations or further clarifications?

Here are five related questions that could expand your understanding:
1. How does the population change at earlier time points, such as \( t = 6 \)?
2. What does the function's behavior suggest about long-term population growth?
3. How would you find when the population reaches a certain size, say 50,000 fish?
4. Can you find the time at which the growth rate \( P'(t) \) reaches a specific value, like 15,000 fish per month?
5. How would changing the initial coefficient of the exponential function affect the population growth?

**Tip:** When dealing with exponential growth, the rate of change is proportional to the population size, which leads to accelerating growth over time.

A fish population is approximated by P(t) = 15e^(0.6t), where t is in months. Calculate and interpret P(12) and P'(12).

Learn how to calculate the population of fish after 12 months and the rate of change in population using the exponential growth model P(t) = 15e^(0.6t). Explore differentiation techniques and exponential functions, suitable for students in Grades 10-12.

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