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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 10,100.

(a) Find a function that models the population [tex] t [/tex] years after 2000 ([tex] t = 0 [/tex] for the year 2000). Hint: Use an exponential function with base [tex] e [/tex].

Your answer is [tex] P(t) = \square [/tex]

(b) Use the function from part (a) to estimate the fox population in the year 2008.

Your answer is (the answer must be an integer) [tex] \square [/tex]



Answer :

Let's break this problem into two parts as per the question.

### Part (a) - Finding the model function for population growth

Given:
- The initial population in the year 2000, [tex]\( P_0 = 10100 \)[/tex]
- Continuous growth rate, [tex]\( r = 0.07 \)[/tex] (7 percent per year)

We need to find a function [tex]\( P(t) \)[/tex] that models the population [tex]\( t \)[/tex] years after 2000.

The formula for exponential growth is:
[tex]\[ P(t) = P_0 \cdot e^{rt} \][/tex]

Plugging in the given values:
[tex]\[ P(t) = 10100 \cdot e^{0.07t} \][/tex]

So, the function that models the population [tex]\( t \)[/tex] years after 2000 is:
[tex]\[ P(t) = 10100 \cdot e^{0.07t} \][/tex]

### Part (b) - Estimating the fox population in the year 2008

To find the population in the year 2008, we need to evaluate the function [tex]\( P(t) \)[/tex] at [tex]\( t = 8 \)[/tex] (since 2008 is 8 years after 2000).

[tex]\[ P(8) = 10100 \cdot e^{0.07 \cdot 8} \][/tex]

Using the provided numerical result:
[tex]\[ P(8) \approx 17681.79225299062 \][/tex]

As the result needs to be an integer:

The estimated fox population in the year 2008 is approximately:
[tex]\[ \boxed{17682} \][/tex]