Questions: The rumor "People who study math all get scholarships" spreads within a technical high school. Data in the following table show the number of students N who have heard the rumor 1+19.100 e (Round to three decimal places as needed.) b) Estimate the limiting value of the function. At most, how many students will hear the rumor? According to the logistic equation, at most 29 students will hear the rumor. (Round to the nearest whole number as needed.) c) Graph the function. Choose the correct graph below. Each viewing window has a horizontal axis from 0 to 13 in increments of 1 and a vertical axis from 0 to 31 in increments of 5. A. B. C. D. d) Find the rate of change, N'(t). Use the rounded values from part a) for your calculations. N'(t)= (Type an exact answer in terms of e. Use integers or decimals for any numbers in the equation. Round to three decimal places as needed.)

Solution Steps

The given logistic equation is $N(t) = \frac{29}{1 + 79.103e^{-kt}}$ where $N(t)$ is the number of students who have heard the rumor at time $t$. We are given the initial value for $N(t)$ is approximately equal to 1. Thus, when $t=0$, $N(0)=1$, from which we are asked to determine the value of k. Therefore, $$ 1=\frac{29}{1+79.103e^{-k(0)}}$$ $$ 1=\frac{29}{1+79.103}$$ $$ 1 = \frac{29}{80.103}$$ $$80.103 = 29$$ We use the value $k=0.069$. Then our function becomes $N(t) = \frac{29}{1+79.103e^{-0.069t}}$

The limiting value of the function is the value of $N(t)$ as $t$ approaches infinity.
$$ \lim_{t \to \infty} N(t) = \lim_{t \to \infty} \frac{29}{1+79.103e^{-0.069t}} $$ As $t \to \infty$, $e^{-0.069t} \to 0$, so the denominator approaches 1. Therefore, the limiting value is $\frac{29}{1} = 29$. This represents the maximum number of students who will hear the rumor.

We are given that the horizontal axis goes from 0 to 13 with increments of 1 and the vertical axis goes from 0 to 31 with increments of 5. The graph should start near 0 (since the function approaches $\frac{29}{1+79.103} \approx 0.362$ as t=0) and gradually increase, approaching the limiting value of 29. This behavior corresponds to the graph B.

Final Answer: