Chapter 5.4, Problem 9E

Spread of News A news item is spread by word of mouth to a potential audience of $10,000$ people. After $t$ days,

$f\left(t\right)=\frac{10,000}{1+50e^{-0.4t}}$

people will have heard the news. The graph of this function is shown in Fig. 7.

Figure 7

a. Approximately how many people will have heard the news after $7$ days?

b. At approximately what rate will the news spread after $14$ days?

c. Approximately when will $7000$ people have heard the news?

d. Approximately when will the news spread at the rate of $600$ people per day?

e. When will the news spread at the greatest rate?

f. Use equations (9) and (10) to determine the differential equation satisfied by $f\left(t\right)$.

g. At what rate will the news spread when half the potential audience has heard the news?