Question
The logistic growth function f left parenthesis t right parenthesis equals StartFraction 105 comma 000 Over 1 plus 4400 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.)
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.)
Answers
GPT-4o mini
To determine the number of people initially infected when the epidemic began (at \( t = 0 \)), we will evaluate the logistic growth function at \( t = 0 \):
\[
f(t) = \frac{105,000}{1 + 4400 e^{-t}}
\]
Substituting \( t = 0 \) into the function:
\[
f(0) = \frac{105,000}{1 + 4400 e^{0}} = \frac{105,000}{1 + 4400 \cdot 1} = \frac{105,000}{1 + 4400} = \frac{105,000}{4401}
\]
Now we will calculate \( \frac{105,000}{4401} \):
\[
f(0) \approx 23.8
\]
Rounding to the nearest whole number, the number of people initially infected is:
\[
\text{Approximately } 24.
\]
So the answer for part (a) is:
**24 people**
(rounded to the nearest whole number as needed).
#### Part 2: How many people were ill by the end of the fourth week?
To find the number of people ill after 4 weeks, we calculate \( f(4) \):
\[
f(4) = \frac{105,000}{1 + 4400 e^{-4}}
\]
First, calculating \( e^{-4} \):
\[
e^{-4} \approx 0.0183156
\]
Now substituting this value back into the function:
\[
f(4) = \frac{105,000}{1 + 4400 \cdot 0.0183156} = \frac{105,000}{1 + 80.38704}
\]
\[
f(4) = \frac{105,000}{81.38704} \approx 1289.669
\]
Rounding to the nearest whole number gives us:
**1290 people** (ill by the end of the fourth week).
#### Part 3: What is the limiting size of the population that becomes ill?
The limiting size of the population that can become ill is given by the numerator of the logistic function. In this case, the maximum possible number of people who can become ill is:
\[
\text{Limiting size} = 105,000
\]
### Summary of Results:
a. The number of people initially infected is **24**.
b. The number of people ill by the end of the fourth week is **1290**.
c. The limiting size of the population that becomes ill is **105,000**.
\[
f(t) = \frac{105,000}{1 + 4400 e^{-t}}
\]
Substituting \( t = 0 \) into the function:
\[
f(0) = \frac{105,000}{1 + 4400 e^{0}} = \frac{105,000}{1 + 4400 \cdot 1} = \frac{105,000}{1 + 4400} = \frac{105,000}{4401}
\]
Now we will calculate \( \frac{105,000}{4401} \):
\[
f(0) \approx 23.8
\]
Rounding to the nearest whole number, the number of people initially infected is:
\[
\text{Approximately } 24.
\]
So the answer for part (a) is:
**24 people**
(rounded to the nearest whole number as needed).
#### Part 2: How many people were ill by the end of the fourth week?
To find the number of people ill after 4 weeks, we calculate \( f(4) \):
\[
f(4) = \frac{105,000}{1 + 4400 e^{-4}}
\]
First, calculating \( e^{-4} \):
\[
e^{-4} \approx 0.0183156
\]
Now substituting this value back into the function:
\[
f(4) = \frac{105,000}{1 + 4400 \cdot 0.0183156} = \frac{105,000}{1 + 80.38704}
\]
\[
f(4) = \frac{105,000}{81.38704} \approx 1289.669
\]
Rounding to the nearest whole number gives us:
**1290 people** (ill by the end of the fourth week).
#### Part 3: What is the limiting size of the population that becomes ill?
The limiting size of the population that can become ill is given by the numerator of the logistic function. In this case, the maximum possible number of people who can become ill is:
\[
\text{Limiting size} = 105,000
\]
### Summary of Results:
a. The number of people initially infected is **24**.
b. The number of people ill by the end of the fourth week is **1290**.
c. The limiting size of the population that becomes ill is **105,000**.