Questions: The fox population in a certain region has a continuous growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 20100. (a) Find a function that models the population t years after 2000 ( t=0 for 2000). Your answer is P(t)=

The fox population in a certain region has a continuous growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 20100. (a) Find a function that models the population t years after 2000 ( t=0 for 2000). Your answer is P(t)=
Transcript text: The fox population in a certain region has a continuous growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 20100. (a) Find a function that models the population $t$ years after 2000 ( $t=0$ for 2000). Your answer is $P(t)=$ $\square$
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Solution

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Solution Steps

To model the population growth, we can use the exponential growth formula \( P(t) = P_0 \cdot e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years after the initial year. Given that the initial population \( P_0 \) is 20100 and the growth rate \( r \) is 0.04 (4 percent), we can plug these values into the formula to get the function.

Step 1: Define the Population Model

The population \( P(t) \) of the foxes can be modeled using the exponential growth formula:

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

where:

  • \( P_0 = 20100 \) (the initial population in the year 2000),
  • \( r = 0.04 \) (the continuous growth rate),
  • \( t \) is the number of years after 2000.
Step 2: Calculate the Population After 10 Years

To find the population 10 years after 2000, we substitute \( t = 10 \) into the population model:

\[ P(10) = 20100 \cdot e^{0.04 \cdot 10} \]

Calculating \( e^{0.4} \) gives approximately \( 1.4918 \). Thus,

\[ P(10) \approx 20100 \cdot 1.4918 \approx 29985.6764 \]

Final Answer

The population of the foxes 10 years after 2000 is approximately

\[ \boxed{P(10) \approx 29985.6764} \]

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