Question

The fox population in a certain region has a continuous growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 20800.

(a) Find a function that models the population
t
years after 2000 (
t
=
0
for 2000).
Hint: Use an exponential function with base
e
.
Your answer is
P
(
t
)
=
20800·e0.06t syntax error.

(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)

Answer 1

To solve the problem, we'll start with part (a) and then move on to part (b).

### (a) Function that models the population

The population growth can be modeled by the formula for continuous exponential growth:

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

where:
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the time in years since the initial time period.

In this case:
- \( P_0 = 20800 \) (the population in the year 2000),
- \( r = 0.06 \) (6% expressed as a decimal),
- \( t \) is the number of years after 2000.

Thus, the function becomes:

\[
P(t) = 20800 \cdot e^{0.06t}
\]

### (b) Estimate the fox population in the year 2008

To find the population in the year 2008, we need to calculate \( P(t) \) where \( t = 8 \) (since 2008 is 8 years after 2000):

\[
P(8) = 20800 \cdot e^{0.06 \cdot 8}
\]

Calculating the exponent:

\[
0.06 \cdot 8 = 0.48
\]

Now we compute \( P(8) \):

\[
P(8) = 20800 \cdot e^{0.48}
\]

Using a calculator to find \( e^{0.48} \):

\[
e^{0.48} \approx 1.615
\]

Now substituting back into the population equation:

\[
P(8) = 20800 \cdot 1.615 \approx 33632
\]

Since we need to round to the nearest integer, the estimated fox population in 2008 is approximately **33632**.

Thus, the final answers are:
- (a) \( P(t) = 20800 \cdot e^{0.06t} \)
- (b) The estimated fox population in 2008 is **33632**.