Let's analyze the given statements based on the function provided:
[tex]\[
N(t) = \frac{300}{1 + 299 e^{-0.36t}}
\][/tex]
### Statement A
It will take approximately 14 minutes for 100 people to hear the rumor.
To verify this statement, we need to evaluate [tex]\( N(t) \)[/tex] at [tex]\( t = 14 \)[/tex] and check if the result is approximately 100.
Given that when we evaluated [tex]\( N(14) \)[/tex], the value was not close to 100, we conclude that this statement is false.
### Statement B
The rumor spreads at a constant rate of 0.36 people per minute.
The function [tex]\( N(t) \)[/tex] is not linear, as it involves an exponential term. A constant rate would imply a linear relationship between [tex]\( t \)[/tex] and [tex]\( N(t) \)[/tex]. Therefore, the rumor does not spread at a constant rate. This statement is false.
### Statement C
There are 299 people in the enclosed space.
To determine the total number of people in the space, we examine the behavior of [tex]\( N(t) \)[/tex] as [tex]\( t \)[/tex] approaches infinity. The denominator's exponential term [tex]\( e^{-0.36t} \)[/tex] approaches zero, so:
[tex]\[
N(t) \ \approx \ \frac{300}{1 + 0} = 300
\][/tex]
Thus, the maximum number of people who can hear the rumor is 300, indicating 300 people are in the enclosed space. Therefore, this statement is false because there are 300 people, not 299.
### Statement D
Initially, only one person had heard the rumor.
To check this, we evaluate [tex]\( N(t) \)[/tex] at [tex]\( t = 0 \)[/tex]:
[tex]\[
N(0) = \frac{300}{1 + 299 e^{0}} = \frac{300}{1 + 299} = \frac{300}{300} = 1
\][/tex]
The calculation shows that initially, only one person had heard the rumor. Thus, this statement is true.
Based on the analysis, the true and false classifications for the statements are:
- Statement A: False
- Statement B: False
- Statement C: False
- Statement D: True
