Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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]
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
