Pine Martens are reintroduced to a forest. Initially, 150 animals are released.  Past data shows the population will increase following an exponential model at a rate of 20% per year. Give the discrete exponential model in the form P(x) = I(b^x). Determine the doubling time, and determine when the population will reach 500 animals.

Given the problem, we can follow these steps to solve it:

### Discrete Exponential Model

The exponential growth model can be expressed as:

\[ P(x) = I \cdot b^x \]

where:
- \( P(x) \) is the population at time \( x \) (in years),
- \( I \) is the initial population,
- \( b \) is the growth factor,
- \( x \) is the number of years.

From the problem, we know:
- The initial population \( I = 150 \),
- The growth rate \( r = 20\% = 0.20 \).

The growth factor \( b \) is calculated as:

\[ b = 1 + r = 1 + 0.20 = 1.20 \]

Thus, the discrete exponential model is:

\[ P(x) = 150 \cdot 1.2^x \]

### Doubling Time

The doubling time \( T_d \) can be found using the formula for exponential growth:

\[ 2 = b^{T_d} \]

Taking the natural logarithm of both sides:

\[ \ln(2) = T_d \ln(b) \]

Solving for \( T_d \):

\[ T_d = \frac{\ln(2)}{\ln(1.2)} \]

Calculating this:

\[ T_d = \frac{\ln(2)}{\ln(1.2)} \approx \frac{0.693}{0.182} \approx 3.807 \text{ years} \]

### Time to Reach 500 Animals

We need to solve for \( x \) when \( P(x) = 500 \):

\[ 500 = 150 \cdot 1.2^x \]

Dividing both sides by 150:

\[ \frac{500}{150} = 1.2^x \]

\[ \frac{10}{3} = 1.2^x \]

Taking the natural logarithm of both sides:

\[ \ln\left(\frac{10}{3}\right) = x \ln(1.2) \]

Solving for \( x \):

\[ x = \frac{\ln\left(\frac{10}{3}\right)}{\ln(1.2)} \]

Calculating this:

\[ x = \frac{\ln\left(\frac{10}{3}\right)}{\ln(1.2)} \approx \frac{1.204}{0.182} \approx 6.615 \text{ years} \]

### Summary

1. **Exponential Model**: \( P(x) = 150 \cdot 1.2^x \)
2. **Doubling Time**: \( \approx 3.807 \) years
3. **Time to reach 500 animals**: \( \approx 6.615 \) years

Would you like more details or have any questions about this problem?

Here are five related questions you might ask:
1. How do we derive the growth factor from the growth rate?
2. What is the significance of the natural logarithm in exponential growth problems?
3. How can we interpret the doubling time in a real-world context?
4. Can the exponential growth model be applied to populations with limiting factors?
5. How do we adjust the model if the growth rate changes over time?

**Tip:** When dealing with exponential growth problems, always ensure that the growth rate and time units are consistent for accurate calculations.

Explore an example of exponential growth in population dynamics with an initial release of pine martens into a forest. Understand how to calculate doubling time and predict population size using the discrete exponential model. Suitable for advanced high school and college students.

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