Suppose that poaching reduces the population of an endangered animal by 1​% per year. Further suppose that when the population of this animal falls below 60​, its extinction is inevitable​ (owing to the lack of reproductive options without severe​ in-breeding). If the current population of the animal is 1700​, when will it face​ extinction? Comment on the validity of this exponential model. 
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Part 1
It will take about 
  
333 years for the animal to face extinction.
​(Do not round until the final answer. Then round to the nearest whole number as​ needed.)
Part 2
Comment on the validity of this exponential model.
The model is 
▼ 
 because the initial population is 
▼ 
 and at this rate of​ decaying, the number of years predicted by the model is 
▼ 
 for the population to face extinction due to the lack of reproductive options. 

Let's solve this step-by-step.

### Part 1: Time to Extinction

1. **Formulate the Exponential Decay Model:**
   Since the population decreases by 1% per year, we can represent the population \( P(t) \) at time \( t \) (in years) using the formula:
   \[
   P(t) = P_0 \cdot e^{-kt}
   \]
   where \( P_0 \) is the initial population and \( k \) is the decay constant.

2. **Define the Initial Conditions and Constants:**
   - Initial population, \( P_0 = 1700 \)
   - Population threshold for extinction, \( P(t) = 60 \)
   - The decay rate is 1% per year, so \( k = 0.01 \).

3. **Set up the Equation and Solve for \( t \):**
   We need to find \( t \) when \( P(t) = 60 \).

   Substitute into the formula:
   \[
   60 = 1700 \cdot e^{-0.01t}
   \]

4. **Solve for \( t \):**
   Divide both sides by 1700:
   \[
   \frac{60}{1700} = e^{-0.01t}
   \]
   Simplify:
   \[
   0.0353 = e^{-0.01t}
   \]
   Take the natural logarithm of both sides:
   \[
   \ln(0.0353) = -0.01t
   \]
   Solve for \( t \):
   \[
   t = \frac{\ln(0.0353)}{-0.01} \approx 332.78
   \]
   Rounding to the nearest whole number, it will take approximately **333 years** for the population to fall below 60, meaning the animal will face extinction in **333 years**.

### Part 2: Validity of the Exponential Model

The validity of this exponential decay model can be questioned for several reasons:

- **Assumptions of Constant Decay Rate:** The model assumes a constant 1% decline each year, which may not accurately reflect real-world population dynamics. Various factors, like conservation efforts or changing environmental conditions, could affect the rate of decline.
  
- **Threshold Effects:** As the population nears 60, other factors such as inbreeding, limited genetic diversity, and difficulty in finding mates would likely accelerate the population decline, deviating from the simple exponential model.

- **Environmental and Biological Factors:** Natural populations don't typically follow a purely exponential model due to complex interactions with the environment, predation, disease, and competition.

In summary, while the exponential model provides a rough estimate, it may not be entirely reliable due to these simplifications.

### Summary of Part 2 Answer:

The model is **questionable** because the initial population is **high**, and at this rate of decay, the number of years predicted (333 years) is **long** for a population that will likely experience additional pressures accelerating extinction due to reproductive challenges and other biological factors.

---

Would you like further details, or have any other questions?

### Related Questions

1. What factors could change the 1% decay rate assumption in real-world scenarios?
2. How would conservation efforts impact this model's validity?
3. What adjustments could be made to model extinction more accurately with smaller populations?
4. How does genetic diversity affect population sustainability at low levels?
5. Can exponential decay models be modified to account for carrying capacities?

### Tip:
In population modeling, it’s useful to consider logistic growth or decline models, which account for factors that limit population expansion or survival.

Suppose that poaching reduces the population of an endangered animal by 1% per year. Further suppose that when the population of this animal falls below 60, its extinction is inevitable (owing to the lack of reproductive options without severe in-breeding). If the current population of the animal is 1700, when will it face extinction? Comment on the validity of this exponential model.

Learn to calculate the time until extinction for an endangered animal population declining by 1% per year, using exponential decay models. Explore the validity of this model, considering biological and environmental factors that affect population dynamics. Suitable for high school students in Grades 10-12.

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