A Population of Foxes A breeding group of foxes is introduced into a protected area, and the population growth follows a logistic pattern. After t years, the population of foxes is given by
N= 37.5 divide 0.25 plus 0.76 exponent t foxes. 

a. How many foxes were introduced into the protected area?

b. Make a graph of N versus t and explain in words how the population of foxes increases with time.

c. When will the fox population reach 100 individuals?

Solve b only 

The graph above shows the population of foxes over time according to the function \( N(t) = \frac{37.5}{0.25 + 0.76^t} \). Here's a breakdown of how the population changes:

- **Initial slow growth**: At the beginning, the population increases slowly because the fox population is still small.
- **Rapid growth phase**: After some time, the population grows rapidly as the conditions are favorable for breeding.
- **Leveling off**: Eventually, the population approaches a maximum or carrying capacity (around 150 individuals), where the growth rate slows down.

Do you have any questions, or would you like more details?

Here are 5 related questions:
1. What is the population after 5 years?
2. How would the graph change if the growth rate \(0.76^t\) were different?
3. Can you calculate the carrying capacity of the fox population from this model?
4. How long does it take for the population to reach half of its carrying capacity?
5. How would this model differ for other animal populations?

**Tip**: In logistic growth models, the population tends to stabilize near the carrying capacity, determined by environmental limitations.

A Population of Foxes A breeding group of foxes is introduced into a protected area, and the population growth follows a logistic pattern. After t years, the population of foxes is given by N = 37.5 / (0.25 + 0.76^t) foxes. 

b. Make a graph of N versus t and explain in words how the population of foxes increases with time.

Explore how the population of foxes grows over time using the logistic growth model N(t) = 37.5 / (0.25 + 0.76^t). This problem provides insight into exponential growth, rapid population increase, and eventual stabilization near the carrying capacity. Suitable for Grades 11-12.

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