To solve this problem, we need to use the exponential decay formula:

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

Where:

• $P(t)$ is the population after time $t$,
• $P_0$ is the initial population,
• $k$ is the decay rate (9% = 0.09),
• $t$ is the time in years,
• $P(t) = 40$ (extinction threshold).

Given Values:

• $P_0 = 1400$,
• $k = 0.09$,
• $P(t) = 40$.

We solve for $t$:

$40 = 1400 \cdot e^{-0.09t}$

Divide both sides by $1400$:

$\frac{40}{1400} = e^{-0.09t}$

Simplify:

$0.02857 \approx e^{-0.09t}$

Take the natural logarithm (ln) of both sides:

$\ln(0.02857) = -0.09t$

$t = \frac{\ln(0.02857)}{-0.09}$

Using a calculator:

$\ln(0.02857) \approx -3.554$

$t = \frac{-3.554}{-0.09} \approx 39.49 \text{ years.}$

Round to the nearest whole number:

$t \approx 39 \text{ years.}$

Final Answer:

It will take approximately 39 years for the animal to face extinction.

Comment on Model Validity:

• The model is valid in terms of calculating population decay due to poaching because it uses exponential decay, which accurately represents the constant proportional decrease.
• However, it may not fully represent real-world scenarios since factors like conservation efforts or environmental changes are not included. Additionally, small populations might face accelerated extinction due to inbreeding or social factors not modeled by exponential decay.

Let me know if you'd like further clarifications!

Questions for Deeper Understanding:

1. What would the extinction time be if the decay rate increased to 10%?
2. How does exponential decay differ from linear decay in population studies?
3. What happens to the model if the extinction threshold changes to a different value?
4. How would conservation efforts (e.g., introducing new animals) affect the validity of the model?
5. Can you derive a generalized formula for extinction time given different decay rates and thresholds?

Tip:

When working with exponential models, always check the assumptions to ensure they match real-world conditions.