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.

Question

(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
)
=

Answer 1

To model the fox population, we need to use an exponential growth function. The formula for continuous exponential growth is given by:

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

where:

• \( P(t) \) is the population at time \( t \),
• \( P_0 \) is the initial population,
• \( r \) is the continuous growth rate,
• \( t \) is the time in years,
• \( e \) is the base of the natural logarithm (approximately equal to 2.71828).

Given:

• The initial population in the year 2000 (\( t = 0 \)) is \( P_0 = 20800 \).
• The continuous growth rate is \( r = 0.06 \) (6 percent expressed as a decimal).

Now we can substitute these values into the formula:

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

So, the function that models the population \( t \) years after 2000 is:

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