The function $N(t)=\frac{300}{1+299 e^{-0.36 t}}$ describes the spread of a rumor among a group of people in an enclosed space. $N$ represents the number of people who have heard the rumor, and $t$ is measured in minutes since the rumor was started. Which of the following statements are true?
Check all that apply.
A. There are 300 people in the enclosed space.
B. Initially, only one person had heard the rumor.
C. The rumor spreads at a constant rate of 0.36 people per minute.
D. It will take 30 minutes for 100 people to hear the rumor.
Answer (2)
Determine the limit of N ( t ) as t approaches infinity: lim t → ∞ N ( t ) = 300 , confirming statement A.
Evaluate N ( 0 ) to find the initial number of people who heard the rumor: N ( 0 ) = 1 , confirming statement B.
Calculate the derivative N ′ ( t ) and observe that it is not constant, disproving statement C.
Solve for t when N ( t ) = 100 , resulting in t ≈ 13.91 , disproving statement D.
The true statements are A and B, so the answer is A and B .
Explanation
Problem Analysis We are given the function N ( t ) = 1 + 299 e − 0.36 t 300 which models the spread of a rumor. We need to determine which of the given statements are true. Let's analyze each statement individually.
Analyzing Statement A Statement A: There are 300 people in the enclosed space. This statement suggests that as time goes to infinity, the number of people who have heard the rumor approaches 300. To check this, we need to find the limit of N ( t ) as t approaches infinity:
t → ∞ lim N ( t ) = t → ∞ lim 1 + 299 e − 0.36 t 300
As t → ∞ , e − 0.36 t → 0 , so the limit becomes:
t → ∞ lim 1 + 299 ( 0 ) 300 = 1 + 0 300 = 300
Thus, statement A is true.
Analyzing Statement B Statement B: Initially, only one person had heard the rumor. To check this, we need to find the value of N ( 0 ) :
N ( 0 ) = 1 + 299 e − 0.36 ( 0 ) 300 = 1 + 299 ( 1 ) 300 = 300 300 = 1
Thus, statement B is true.
Analyzing Statement C Statement C: The rumor spreads at a constant rate of 0.36 people per minute. To check this, we need to find the derivative of N ( t ) with respect to t :
N ′ ( t ) = d t d ( 1 + 299 e − 0.36 t 300 )
Using the quotient rule or chain rule, we find:
N ′ ( t ) = ( 1 + 299 e − 0.36 t ) 2 300 ⋅ ( 299 ⋅ 0.36 e − 0.36 t ) = ( 1 + 299 e − 0.36 t ) 2 32292 e − 0.36 t
Since N ′ ( t ) is not constant, the rumor does not spread at a constant rate. Thus, statement C is false.
Analyzing Statement D Statement D: It will take 30 minutes for 100 people to hear the rumor. To check this, we need to solve for t when N ( t ) = 100 :
100 = 1 + 299 e − 0.36 t 300
1 + 299 e − 0.36 t = 3
299 e − 0.36 t = 2
e − 0.36 t = 299 2
− 0.36 t = ln ( 299 2 )
t = − 0.36 ln ( 299 2 ) = − 0.36 ln ( 2 ) − ln ( 299 ) ≈ 13.91
Since t ≈ 13.91 minutes, statement D is false.
Conclusion Based on our analysis, statements A and B are true, while statements C and D are false.
Examples
Logistic functions, like the one describing the rumor spread, are incredibly useful in modeling various real-world phenomena. For instance, they can describe the growth of a population within a limited environment, the adoption rate of a new technology, or even the spread of a disease. Understanding these models helps us predict and manage these phenomena effectively, whether it's planning resource allocation, marketing strategies, or public health interventions. The key is that logistic functions capture the idea of initial exponential growth that slows down as it approaches a carrying capacity or saturation point.
Statements A and B are true: there are 300 people in the space and initially, only one person had heard the rumor. Statements C and D are false. Thus, the correct answers are A and B.
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