Question
The logistic growth function f left parenthesis t right parenthesis equals StartFraction 105 comma 000 Over 1 plus 4500 e Superscript negative t EndFraction
describes the number of people, f(t), who have become ill with influenza t weeks after its initial outbreak in a particular community.
a. How many people became ill with the flu when the epidemic began?
b. How many people were ill by the end of the fourth week?
c. What is the limiting size of the population that becomes ill?
Question content area bottom
Part 1
a. The number of people initially infected is
enter your response here.
(Round to the nearest whole number as needed.)
Part 2
b. The number of people infected after 4 weeks is
enter your response here.
(Round to the nearest whole number as needed.)
Part 3
c. The limiting size of the infected population is
enter your response here.
(Round to the nearest whole number as needed.)
describes the number of people, f(t), who have become ill with influenza t weeks after its initial outbreak in a particular community.
a. How many people became ill with the flu when the epidemic began?
b. How many people were ill by the end of the fourth week?
c. What is the limiting size of the population that becomes ill?
Question content area bottom
Part 1
a. The number of people initially infected is
enter your response here.
(Round to the nearest whole number as needed.)
Part 2
b. The number of people infected after 4 weeks is
enter your response here.
(Round to the nearest whole number as needed.)
Part 3
c. The limiting size of the infected population is
enter your response here.
(Round to the nearest whole number as needed.)
Answers
GPT-4o mini
To evaluate the logistic growth function \( f(t) = \frac{105,000}{1 + 4500e^{-t}} \), we can answer the questions step by step:
### Part 1: Initial Number of People Infected \( (t = 0) \)
To find how many people became ill at the epidemic's onset, we substitute \( t = 0 \) into the function:
\[
f(0) = \frac{105,000}{1 + 4500e^{0}} = \frac{105,000}{1 + 4500} = \frac{105,000}{4501}
\]
Now we calculate:
\[
f(0) = \frac{105,000}{4501} \approx 23.3
\]
Rounding to the nearest whole number gives:
- **Initial number of infected people:** \(\approx 23\)
### Part 2: Number of People Infected After 4 Weeks \( (t = 4) \)
Now, to find the number of people infected after 4 weeks, we substitute \( t = 4 \) into the function:
\[
f(4) = \frac{105,000}{1 + 4500e^{-4}}
\]
Calculating \( e^{-4} \):
\[
e^{-4} \approx 0.0183
\]
Now substitute back into the function:
\[
f(4) = \frac{105,000}{1 + 4500 \times 0.0183} = \frac{105,000}{1 + 82.35} = \frac{105,000}{83.35}
\]
Now, calculate:
\[
f(4) \approx \frac{105,000}{83.35} \approx 1258.5
\]
Rounding to the nearest whole number gives:
- **Number of people infected after 4 weeks:** \(\approx 1259\)
### Part 3: Limiting Size of the Infected Population
The limiting size of the population that becomes ill corresponds to the function's maximum, which is the numerator in the logistic function:
- **Limiting size is:** \( 105,000 \)
### Summary of Results:
a. The number of people initially infected is **23**.
b. The number of people infected after 4 weeks is **1259**.
c. The limiting size of the infected population is **105000**.
### Part 1: Initial Number of People Infected \( (t = 0) \)
To find how many people became ill at the epidemic's onset, we substitute \( t = 0 \) into the function:
\[
f(0) = \frac{105,000}{1 + 4500e^{0}} = \frac{105,000}{1 + 4500} = \frac{105,000}{4501}
\]
Now we calculate:
\[
f(0) = \frac{105,000}{4501} \approx 23.3
\]
Rounding to the nearest whole number gives:
- **Initial number of infected people:** \(\approx 23\)
### Part 2: Number of People Infected After 4 Weeks \( (t = 4) \)
Now, to find the number of people infected after 4 weeks, we substitute \( t = 4 \) into the function:
\[
f(4) = \frac{105,000}{1 + 4500e^{-4}}
\]
Calculating \( e^{-4} \):
\[
e^{-4} \approx 0.0183
\]
Now substitute back into the function:
\[
f(4) = \frac{105,000}{1 + 4500 \times 0.0183} = \frac{105,000}{1 + 82.35} = \frac{105,000}{83.35}
\]
Now, calculate:
\[
f(4) \approx \frac{105,000}{83.35} \approx 1258.5
\]
Rounding to the nearest whole number gives:
- **Number of people infected after 4 weeks:** \(\approx 1259\)
### Part 3: Limiting Size of the Infected Population
The limiting size of the population that becomes ill corresponds to the function's maximum, which is the numerator in the logistic function:
- **Limiting size is:** \( 105,000 \)
### Summary of Results:
a. The number of people initially infected is **23**.
b. The number of people infected after 4 weeks is **1259**.
c. The limiting size of the infected population is **105000**.