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.