The current population of a threatened animal species is 

1.31.3 

​million, but it is declining with a​ half-life of 

2020 

years. How many animals will be left in 

3030 

​years?

To determine the population of the threatened animal species after 30 years, we can use the formula for exponential decay, which is analogous to exponential growth but for a declining population:

\[
P(t) = P_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}
\]

where:
- \( P(t) \) is the population after time \( t \).
- \( P_0 \) is the initial population.
- \( t \) is the time elapsed.
- \( T \) is the half-life.

Given:
- \( P_0 = 1.3 \) million
- \( t = 30 \) years
- \( T = 20 \) years

Substituting the values into the formula:

\[
P(30) = 1.3 \times \left(\frac{1}{2}\right)^{\frac{30}{20}} = 1.3 \times \left(\frac{1}{2}\right)^{1.5}
\]

Now, calculate \( \left(\frac{1}{2}\right)^{1.5} \):

\[
\left(\frac{1}{2}\right)^{1.5} = \frac{1}{2^{1.5}} = \frac{1}{\sqrt{2^3}} = \frac{1}{\sqrt{8}} \approx 0.3536
\]

Thus,

\[
P(30) \approx 1.3 \times 0.3536 \approx 0.4597 \text{ million}
\]

So, after 30 years, approximately **0.46 million** animals (or 460,000 animals) will be left.

Would you like more details or have any questions? Here are some related questions:

1. What will the population be after 50 years?
2. How does changing the half-life affect the population decay rate?
3. If the initial population was 2 million instead of 1.3 million, what would be the population after 30 years?
4. How does exponential decay differ from exponential growth in real-world scenarios?
5. Can exponential decay apply to non-population scenarios, such as radioactive decay?

**Tip:** In exponential decay, the population never truly reaches zero but continues to halve at regular intervals based on the half-life.

Learn how to calculate the population decline of a threatened animal species using exponential decay. Given an initial population of 1.3 million and a half-life of 20 years, find out how many animals will be left after 30 years. Explore related questions about exponential decay and population dynamics.

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