A biologist observes a population of birds over a 12 year period. At the beginning of the period , there were 257 thousand birds. By the end of the period , there were 179 thousand birds.

Assume the population of birds can be modeled using an exponential function.

If 
 is the population (in thousands) , where 
 is the time since the start of the period, then 
Incorrect   (Round your parameters to four decimal places)

We are given that the bird population follows an exponential decay function. The general form for an exponential decay model is:

\[
P(t) = P_0 e^{kt}
\]

Where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population,
- \(k\) is the decay rate (which will be negative in this case since the population is decreasing),
- \(t\) is the time in years.

### Given:
- Initial population \(P_0 = 257\) (in thousands),
- Population after 12 years \(P(12) = 179\),
- Time period \(t = 12\).

### Step 1: Use the population formula at \(t = 12\)
\[
P(12) = P_0 e^{12k}
\]

Substitute the known values:
\[
179 = 257 e^{12k}
\]

### Step 2: Solve for \(k\)
First, isolate the exponential term by dividing both sides by 257:
\[
\frac{179}{257} = e^{12k}
\]

Take the natural logarithm of both sides to eliminate the exponential:
\[
\ln\left(\frac{179}{257}\right) = 12k
\]

Now solve for \(k\):
\[
k = \frac{\ln\left(\frac{179}{257}\right)}{12}
\]

### Step 3: Calculate the value of \(k\)
Let's compute \(k\).The decay rate \(k\) is approximately \(-0.0301\).

### Step 4: Write the exponential model
Now that we have the value of \(k\), the exponential model for the bird population over time is:

\[
P(t) = 257 e^{-0.0301t}
\]

Would you like further clarification, or do you have any questions about this model?

Here are five related questions to explore:
1. How would the population change after 5 years?
2. What would the population be after 20 years?
3. How does the rate of decay compare if we adjust the model for a faster decline?
4. Can we use this model to predict when the population will reach 100 thousand?
5. How does exponential decay differ from linear decay in population models?

**Tip:** In exponential decay models, the decay rate \(k\) determines how quickly the population decreases—small changes in \(k\) can lead to significant long-term differences in the population size.

A biologist observes a population of birds over a 12 year period. At the beginning of the period, there were 257 thousand birds. By the end of the period, there were 179 thousand birds. Assume the population of birds can be modeled using an exponential function. If P(t) is the population (in thousands), where t is the time since the start of the period, then (Round your parameters to four decimal places)

Explore how to model a declining bird population over 12 years using exponential decay. The problem involves calculating the decay rate and applying exponential functions to predict population changes. This problem is suitable for students in Grades 10-12.

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