the function n(t) = 300 / (1 + 299e^(-0.36t)) describes the spread of a rumor among a group of people in an…

the function n(t) = 300 / (1 + 299e^(-0.36t)) 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. initially, only one person had heard the rumor. b. the rumor spreads at a constant rate of 0.36 people per minute. c. there are 299 people in the enclosed space. d. it will take approximately 14 minutes for 100 people to hear the rumor.

the function n(t) = 300 / (1 + 299e^(-0.36t)) 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. initially, only one person had heard the rumor. b. the rumor spreads at a constant rate of 0.36 people per minute. c. there are 299 people in the enclosed space. d. it will take approximately 14 minutes for 100 people to hear the rumor.

Answer

Explanation:

Step1: Check option A

When (t = 0), (N(0)=\frac{300}{1 + 299e^{-0.36\times0}}=\frac{300}{1 + 299}=\frac{300}{300}=1). So initially, only one person had heard the rumor.

Step2: Analyze the rate of spread

The function (N(t)=\frac{300}{1 + 299e^{-0.36t}}) is a logistic - growth function. The rate of change is not constant. The derivative of (N(t)) with respect to (t) is (N^\prime(t)=\frac{300\times299\times0.36e^{-0.36t}}{(1 + 299e^{-0.36t})^2}), which is not a constant function. So option B is false.

Step3: Determine the total number of people

As (t\rightarrow\infty), (e^{-0.36t}\rightarrow0), and (N(t)\rightarrow300). So there are 300 people in the enclosed space, not 299. Option C is false.

Step4: Solve for (t) when (N(t)=100)

Set (N(t) = 100), then (100=\frac{300}{1 + 299e^{-0.36t}}). Cross - multiply gives (100(1 + 299e^{-0.36t})=300), (1+299e^{-0.36t}=3), (299e^{-0.36t}=2), (e^{-0.36t}=\frac{2}{299}). Take the natural logarithm of both sides: (-0.36t=\ln(\frac{2}{299})), (t=\frac{\ln(\frac{2}{299})}{- 0.36}\approx\frac{\ln(2)-\ln(299)}{-0.36}\approx\frac{0.693 - 5.704}{-0.36}\approx\frac{-5.011}{-0.36}\approx14).

Answer:

A. Initially, only one person had heard the rumor. D. It will take approximately 14 minutes for 100 people to hear the rumor.