The fox population in a certain region has a continuous growth rate of 7 percent per year. It is estimated that the population in the year 2000 was 8100.

To model the fox population, we can use the formula for continuous exponential growth:

\[
P(t) = P_0 e^{rt}
\]

where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the continuous growth rate, and
- \( t \) is the time in years since the starting year.

Given:
- \( r = 0.07 \) (since 7% growth rate is \(0.07\) as a decimal),
- \( P_0 = 8100 \) (population in the year 2000),
- Year 2000 corresponds to \( t = 0 \).

### Population at Any Future Time
To find the population at any future time \( t \) years after 2000, substitute \( P_0 = 8100 \) and \( r = 0.07 \) into the formula:

\[
P(t) = 8100 \cdot e^{0.07t}
\]

### Example Calculation
If we want to find the population in a specific year, such as 2010, where \( t = 10 \):

\[
P(10) = 8100 \cdot e^{0.07 \times 10}
\]

Calculating \( e^{0.7} \) (approximately 2.0138):

\[
P(10) \approx 8100 \cdot 2.0138 \approx 16311.78
\]

So, in 2010, the fox population would be approximately 16,312.

Would you like additional details or have specific years in mind?

---

**Relative Questions**:
1. How would a change in growth rate affect the population in 2010?
2. How can we calculate the population doubling time with a continuous growth rate?
3. How would the formula change if the growth rate were compounded annually instead of continuously?
4. What would happen to the population if the growth rate were negative?
5. How can we estimate the fox population in 2050?

**Tip**: To approximate future populations, you can use the rule of 70: divide 70 by the growth rate percentage to find the approximate doubling time.

Explore the exponential growth model for the fox population, initially 8100 in 2000 with a continuous growth rate of 7%. Calculate future populations, including an estimate for 2010.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation