Simone is an ecologist who studies the change in the tiger population of Siberia over time.
The relationship between the elapsed time
\[t\], in years, since Simone started studying the population, and the number of tigers,
\[N_{\text{year}}(t)\], is modeled by the following function:

\[N_{\text{year}}(t)=612\cdot \left(\dfrac{2}{3}\right)\^{\Large t}\]
Complete the following sentence about the monthly rate of change in the tiger population.
Round your answer to two decimal places.
Every month, the number of tigers decays by a factor of
.

The given function that models the tiger population is:

\[
N_{\text{year}}(t) = 612 \cdot \left(\frac{2}{3}\right)^t
\]

Here, \( t \) is the time in years, and the base \( \frac{2}{3} \) represents the annual decay factor of the tiger population.

To find the monthly rate of decay, we need to convert the annual decay factor into a monthly decay factor. The annual decay factor is \(\frac{2}{3}\), so we need to raise this to the power of \(\frac{1}{12}\) to find the monthly decay factor:

\[
\text{Monthly decay factor} = \left(\frac{2}{3}\right)^{\frac{1}{12}}
\]

Now, let's calculate this value.

\[
\text{Monthly decay factor} \approx \left(\frac{2}{3}\right)^{\frac{1}{12}} \approx 0.9766
\]

So, every month, the number of tigers decays by a factor of approximately **0.98** when rounded to two decimal places.

Would you like more details or have any questions?

Here are 5 relative questions to expand your understanding:
1. How does the annual decay factor affect the population over multiple years?
2. What would the population be after 3 years according to the given model?
3. How would the model change if the population increased annually instead of decreasing?
4. How can the decay factor be interpreted in a real-world ecological context?
5. What would be the monthly population decay in percentage terms?

**Tip:** When working with exponential decay models, always ensure to convert time units consistently when interpreting the decay factor.

Explore how the tiger population in Siberia decays over time using an exponential decay model. Learn the annual and monthly decay factors, with detailed calculations and interpretations. Suitable for Grades 10-12.

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