Rumor Problem: There are 1000 students attending Lowe High. One day 10 students arrive

Chapter 9, Problem 19

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Rumor Problem: There are 1000 students attending Lowe High. One day 10 students arrive at school bearing the rumor that final exams will be canceled! On average, each student talks to other students at a rate of two students per hour, passing on the rumor to students, some of whom have already heard it and some of whom have not. Thus, the rate at which students hear the rumor for the first time is 2 times the number who have already heard it times the fraction of the students who have not yet heard it. If y is the number of students who have heard the rumor at time t, in hours since school started, then a. Solve this differential equation algebraically, subject to the initial condition that y = 10 when school started at t = 0. b. How many students had heard the rumor after the first hour? At lunchtime (t = 4)? At the end of the school day (t = 8)? c. How many students had heard the rumor at the time it was spreading the fastest? At what time was this? d. Figure 9-7a shows the slope field for the differential equation. Plot the solution in part a on a copy of the figure. How does the curve relate to the slope field? Figure 9-7a