A breeding group of foxes is introduced into a protected area and exhibits logistic population growth. After *t* years the number of foxes is given by

*N*(*t*) = 

37.5

0.25 + 0.76*t*

 foxes.

(a) How many foxes were introduced into the protected area?
  foxes

(b) Calculate *N*(2). (Round your answer to the nearest whole number.)
*N*(2) =  

Explain the meaning of the number you have calculated.

This means that after 2 years there were about  foxes in the protected area.


(c) Explain how the population varies with time. Include in your explanation the average rate of increase over the first 10-year period and the average rate of increase over the second 10-year period.


The fox population is constant.The fox population is growing, but the population does not grow as rapidly in early years as it did later on.     The fox population is growing, but the population does not grow as rapidly in later years as it did early on.


(d) Find the carrying capacity for foxes in the protected area.
  foxes

(e) As we saw in the discussion of terminal velocity for a skydiver, the question of when the carrying capacity is reached may lead to an involved discussion. We ask the question differently. When is 95% of carrying capacity reached? (Round your answer to the nearest year.)
after   years

We are given a logistic growth model for the fox population, described by the function:

\[
N(t) = \frac{37.5}{0.25 + 0.76t}
\]

where \( t \) is time in years, and \( N(t) \) gives the number of foxes at time \( t \). Let's tackle each part of the question:

---

### (a) **How many foxes were introduced into the protected area?**

To find the number of foxes initially introduced, we evaluate \( N(0) \), which represents the population at \( t = 0 \).

\[
N(0) = \frac{37.5}{0.25 + 0.76 \cdot 0} = \frac{37.5}{0.25} = 150
\]

**Answer:** 150 foxes were introduced into the protected area.

---

### (b) **Calculate \( N(2) \).**

To calculate the number of foxes after 2 years, we evaluate \( N(2) \):

\[
N(2) = \frac{37.5}{0.25 + 0.76 \cdot 2} = \frac{37.5}{0.25 + 1.52} = \frac{37.5}{1.77} \approx 21.19
\]

Rounding to the nearest whole number:

\[
N(2) \approx 21 \text{ foxes}
\]

**Explanation:** After 2 years, there were about **21 foxes** in the protected area.

---

### (c) **Explain how the population varies with time.**

The population follows a **logistic growth model**, which typically means that the population grows rapidly at first and then slows as it approaches the carrying capacity (the maximum population that the environment can sustain). 

- In the **early years**, the population grows rapidly.
- As time goes on, the population growth **slows down** and approaches the carrying capacity.

To understand the population's growth rates over specific intervals:

#### Average Rate of Increase (First 10 Years):
We compute \( N(10) \) and compare it with the initial population \( N(0) \).

\[
N(10) = \frac{37.5}{0.25 + 0.76 \cdot 10} = \frac{37.5}{7.85} \approx 4.78
\]

The average rate of increase in the first 10 years is:

\[
\text{Rate (First 10 years)} = \frac{N(10) - N(0)}{10} = \frac{4.78 - 150}{10} \approx -14.52
\]

Thus, in the first 10 years, the population decreases at an average rate of about **14.5 foxes per year**.

---

### (d) **Find the carrying capacity for foxes in the protected area.**

The carrying capacity of a population in a logistic growth model is the horizontal asymptote of the function as \( t \to \infty \). Mathematically, this is the limit of \( N(t) \) as \( t \to \infty \):

\[
\lim_{t \to \infty} N(t) = \lim_{t \to \infty} \frac{37.5}{0.25 + 0.76t} = 0
\]

Thus

A breeding group of foxes is introduced into a protected area and exhibits logistic population growth. After t years the number of foxes is given by N(t) = 37.5 / (0.25 + 0.76 * t) foxes.

(a) How many foxes were introduced into the protected area?
(b) Calculate N(2). (Round your answer to the nearest whole number.)
Explain the meaning of the number you have calculated.
(c) Explain how the population varies with time. Include in your explanation the average rate of increase over the first 10-year period and the average rate of increase over the second 10-year period.
(d) Find the carrying capacity for foxes in the protected area.
(e) When is 95% of carrying capacity reached?

Explore the logistic population growth of foxes in a protected area. Solve for the initial population, fox count after 2 years, and carrying capacity. Learn how the population growth rate changes over time, with a detailed analysis of the logistic growth formula. Ideal for students in Grades 10-12.

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