A rumor spreads through a school. Let y(t) be the fraction of the population that has heard the rumor at time t and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction 1-y that has not yet heard the rumor. That is, assume y(t) = ky(1-y) where k is an unknown constant. The school has 1000 students in total. At 8 a.m., 97 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model, determine how much time passes before 90% of the students will have heard the rumor. (Note: Since y(t) denotes a proportion, 0 ≤ y(t) < 1 for all t, and y = 1 means 100% of the students know.) 90% of the students have heard the rumor after about hours.
A rumor spreads through a school. Let y(t) be the fraction of the population that has heard the rumor at time t and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction 1-y that has not yet heard the rumor. That is, assume y(t) = ky(1-y) where k is an unknown constant. The school has 1000 students in total. At 8 a.m., 97 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model, determine how much time passes before 90% of the students will have heard the rumor. (Note: Since y(t) denotes a proportion, 0 ≤ y(t) < 1 for all t, and y = 1 means 100% of the students know.) 90% of the students have heard the rumor after about hours.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A rumor spreads through a school. Let y(t) be the fraction of the population that has heard the rumor at
time t and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of
the population that has heard the rumor and the fraction 1-y that has not yet heard the rumor. That is,
assume
y(t) = ky(1 - y)
where k is an unknown constant. The school has 1000 students in total. At 8 a.m., 97 students have heard
the rumor, and by noon, half the school has heard it. Using the logistic model, determine how much time
passes before 90% of the students will have heard the rumor. (Note: Since y(t) denotes a proportion,
0 ≤ y(t) ≤ 1 for all t, and y = 1 means 100% of the students know.)
90% of the students have heard the rumor after about
hours.
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