Let's examine each of the statements one by one, using the function [tex]\( N(t) = \frac{300}{1 + 299 e^{-0.36 t}} \)[/tex] that defines the spread of the rumor.
Statement A: The rumor spreads at a constant rate of 0.36 people per minute.
In general, the term [tex]\( e^{-0.36t} \)[/tex] in the given function indicates an exponential decay, not a constant rate of change. The function actually describes a logistic growth, where the rate of spread of the rumor decreases over time as more people hear the rumor.
So, Statement A is False.
Statement B: There are 300 people in the enclosed space.
Let's examine the function. The number 300 is the numerator of the fraction in [tex]\( N(t) \)[/tex]. This suggests that the total number of people who can potentially hear the rumor is capped at 300, indicating the total population in the enclosed space.
So, Statement B is True.
Statement C: It will take 30 minutes for 100 people to hear the rumor.
To test this statement, we need to solve for [tex]\( t \)[/tex] when [tex]\( N(t) = 100 \)[/tex]:
[tex]\[ 100 = \frac{300}{1 + 299 e^{-0.36 t}} \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[
100 (1 + 299 e^{-0.36 t}) = 300 \\
1 + 299 e^{-0.36 t} = 3 \\
299 e^{-0.36 t} = 2 \\
e^{-0.36 t} = \frac{2}{299} \\
-0.36 t = \ln \left( \frac{2}{299} \right) \\
t = \frac{\ln(2/299)}{-0.36}
\][/tex]
After solving this equation numerically, we find that the time [tex]\( t \)[/tex] does not equal 30 minutes. Based on the previous verification, we see that the correct solution yields a different value for [tex]\( t \)[/tex].
So, Statement C is False.
Statement D: Initially, only one person had heard the rumor.
At [tex]\( t = 0 \)[/tex], we find [tex]\( N(0) \)[/tex]:
[tex]\[ N(0) = \frac{300}{1 + 299 e^{-0.36 \cdot 0}} = \frac{300}{1 + 299 \times 1} = \frac{300}{300} = 1 \][/tex]
This shows that initially, at time [tex]\( t = 0 \)[/tex], the number of people who had heard the rumor was indeed 1.
So, Statement D is True.
Conclusions:
- A is False
- B is True
- C is False
- D is True
Thus, the correct answers are:
[tex]\[ \boxed{\text{B and D}} \][/tex]
