The current population of a threatened animal species is 1.3 ​million, but it is declining with a​ half-life of 20 years. How many animals will be left in 30 ​years?

To determine the population of a species after 30 years with a half-life of 20 years, we can use the exponential decay formula:

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

Where:
- \( P(t) \) is the population after time \( t \) (in years),
- \( P_0 \) is the initial population (1.3 million),
- \( T_{\text{half}} \) is the half-life (20 years),
- \( t \) is the elapsed time (30 years).

Now, plugging in the values:

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

We calculate \( \left( \frac{1}{2} \right)^{1.5} \) and then multiply by 1.3.

Let me calculate this for you.After 30 years, the population of the threatened animal species will be approximately 0.46 million, or 460,000 animals.

Would you like more details on this process or have any other questions?

Here are 5 related questions for further exploration:
1. How would the population change if the half-life was shorter, say 10 years?
2. What is the expected population after 50 years with the same half-life?
3. How would we calculate the initial population if we only knew the population after 30 years?
4. How does exponential decay differ from linear population decline?
5. What mathematical techniques are used to estimate population growth in species not facing extinction?

**Tip**: Exponential decay models are often used in biology, physics, and finance to describe processes where quantities decrease rapidly at first and then slow down.

Learn how to calculate the population of a threatened animal species after 30 years using the exponential decay formula. Starting with an initial population of 1.3 million and a half-life of 20 years, discover how the population will decline to approximately 460,000. Suitable for high school students learning exponential decay and half-life concepts.

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