At the beginning of 2000 there were 3455 bears living in a nature reserve.
The population is declining by  each year and after 9 years has reduced to 2828.

Find the value of .

To find the value of the annual percentage decline in the bear population, we can use an exponential decay model. The formula for an exponential decay is given by:

\[
P(t) = P_0 \times (1 - r)^t
\]

Where:
- \( P(t) \) is the population after time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (the value we need to find),
- \( t \) is the number of years.

In this case:
- \( P_0 = 3455 \) (initial population),
- \( P(9) = 2828 \) (population after 9 years),
- \( t = 9 \) (years).

We can now plug these values into the equation:

\[
2828 = 3455 \times (1 - r)^9
\]

First, solve for \( (1 - r)^9 \):

\[
\frac{2828}{3455} = (1 - r)^9
\]

\[
0.8185 \approx (1 - r)^9
\]

Now, take the 9th root of both sides to solve for \( 1 - r \):

\[
(1 - r) = (0.8185)^{\frac{1}{9}}
\]

Using a calculator:

\[
1 - r \approx 0.9791
\]

So:

\[
r \approx 1 - 0.9791 = 0.0209
\]

Thus, the rate of decline is approximately:

\[
r \approx 0.0209 \text{ or } 2.09\%
\]

So, the bear population is declining by about 2.09% per year.

Would you like more details or have any questions?

### Related questions:
1. How would the formula change if the population were increasing?
2. What would be the population after 15 years at the same rate of decline?
3. How long would it take for the population to halve at this rate?
4. What is the continuous decay model for this situation?
5. How would the population change if the decline rate increased to 3%?

**Tip:** In exponential decay problems, small changes in the rate can result in significant long-term effects on the population.

At the beginning of 2000 there were 3455 bears living in a nature reserve. The population is declining by each year and after 9 years has reduced to 2828. Find the value of the annual decline rate.

Learn how to calculate the annual decline rate of a bear population using the exponential decay formula. This problem explores key concepts like exponential decay, logarithms, and algebra. Suitable for Grades 9-12, the solution provides a step-by-step guide to finding the rate of decline from 3455 bears to 2828 over 9 years.

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