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The fox population in a certain region has an annual growth rate of 5 percent per year. It is estimated that the population in the year 2000 was 19,400.

(a) Find a function that models the population \( t \) years after 2000 (\( t = 0 \) for the year 2000).

Your answer is \( P(t) = \ \square \)

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

Your answer is (the answer should be an integer) \( \square \)

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Note: The placeholders ([tex]\(\square\)[/tex]) are meant for students to fill in their answers.



Answer :

GinnyAnswer
Let me guide you through the problem step-by-step:

### Part (a)
We need to find a function that models the population \( P(t) \), where \( t \) is the number of years after 2000.

Given:
- The initial population in the year 2000 (which we consider as \( t = 0 \)) is \( 19400 \).
- The annual growth rate is 5%.

Since the fox population grows exponentially, we can use the formula for exponential growth:
[tex]\[ P(t) = P_0 \times (1 + r)^t \][/tex]

where:
- \( P(t) \) is the population after \( t \) years.
- \( P_0 \) is the initial population.
- \( r \) is the growth rate.
- \( t \) is the time in years after the initial time.

Substituting the given values:
- \( P_0 = 19400 \)
- \( r = 0.05 \)

The function that models the population \( t \) years after 2000 is:
[tex]\[ P(t) = 19400 \times (1 + 0.05)^t \][/tex]

### Part (b)
Now, we need to estimate the fox population in the year 2008.

First, calculate \( t \) for the year 2008:
[tex]\[ t = 2008 - 2000 = 8 \][/tex]

Next, use the function found in part (a):
[tex]\[ P(t) = 19400 \times (1 + 0.05)^t \][/tex]

Substitute \( t = 8 \) into the equation:
[tex]\[ P(8) = 19400 \times (1 + 0.05)^8 \][/tex]

After evaluating the function, we get the population in the year 2008:
[tex]\[ P(8) \approx 28663 \][/tex]

So, the estimated fox population in the year 2008 is approximately:
[tex]\[ \boxed{28663} \][/tex]

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