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 29700. (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) = (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)
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 29700. (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) = (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)
Trigonometry (MindTap Course List)
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Chapter5: Exponential And Logarithmic Functions
Section5.5: Exponential And Logarithmic Models
Problem 30E: Population The table shows the mid-year populations (in millions) of five countries in 2015 and the...
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![### Modeling Fox Population Growth
In this exercise, we will model the fox population in a certain region given a continuous growth rate and an initial population. The objective is to find an exponential function that describes this growth over time and use it to estimate future populations.
#### Given Data:
- The fox population in the year 2000 (t = 0) was 29,700.
- The population has a continuous growth rate of 4 percent per year.
#### Tasks:
1. **Find a function** that models the population \( t \) years after 2000 (where \( t = 0 \) for the year 2000). Use an exponential function with the base \( e \).
2. **Estimate the fox population** in the year 2008 using the function from part (a).
### Solution:
#### (a) Finding the Function
The general form of an exponential function for continuous growth is:
\[ P(t) = P_0 e^{rt} \]
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 since the initial time
Given:
- \( P_0 = 29700 \)
- \( r = 0.04 \) (4 percent per year)
Substituting the given values:
\[ P(t) = 29700 \cdot e^{0.04t} \]
#### (b) Estimating the Population in 2008
The year 2008 is 8 years after 2000, hence \( t = 8 \).
Using the function derived in part (a):
\[ P(8) = 29700 \cdot e^{0.04 \cdot 8} \]
Calculate \( e^{0.32} \):
\[ e^{0.32} \approx 1.377127764 \]
Next, multiply by the initial population:
\[ P(8) \approx 29700 \cdot 1.377127764 \approx 40903.1 \]
Since the population must be an integer, we round to the nearest integer:
\[ P(8) \approx 40903 \]
Thus, the estimated fox population in the year 2008 is 40,903.
### Summary
**Function derived:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d8b0e94-47ef-477e-9622-abc8b98247d8%2Ffffb5775-44a9-4b69-a9c4-b451f67a0bb7%2Ftvt8tp9_processed.png&w=3840&q=75)
Transcribed Image Text:### Modeling Fox Population Growth
In this exercise, we will model the fox population in a certain region given a continuous growth rate and an initial population. The objective is to find an exponential function that describes this growth over time and use it to estimate future populations.
#### Given Data:
- The fox population in the year 2000 (t = 0) was 29,700.
- The population has a continuous growth rate of 4 percent per year.
#### Tasks:
1. **Find a function** that models the population \( t \) years after 2000 (where \( t = 0 \) for the year 2000). Use an exponential function with the base \( e \).
2. **Estimate the fox population** in the year 2008 using the function from part (a).
### Solution:
#### (a) Finding the Function
The general form of an exponential function for continuous growth is:
\[ P(t) = P_0 e^{rt} \]
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 since the initial time
Given:
- \( P_0 = 29700 \)
- \( r = 0.04 \) (4 percent per year)
Substituting the given values:
\[ P(t) = 29700 \cdot e^{0.04t} \]
#### (b) Estimating the Population in 2008
The year 2008 is 8 years after 2000, hence \( t = 8 \).
Using the function derived in part (a):
\[ P(8) = 29700 \cdot e^{0.04 \cdot 8} \]
Calculate \( e^{0.32} \):
\[ e^{0.32} \approx 1.377127764 \]
Next, multiply by the initial population:
\[ P(8) \approx 29700 \cdot 1.377127764 \approx 40903.1 \]
Since the population must be an integer, we round to the nearest integer:
\[ P(8) \approx 40903 \]
Thus, the estimated fox population in the year 2008 is 40,903.
### Summary
**Function derived:
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