Homework Sourse1 pt Suppose that news spreads through a city of fixed size
(1 pt) Suppose that news spreads through a city of fixed size of 900000 people at a time rate proportional to the number of people who have not heard the news.
Let y(t) denote the number of people who have heard the news t days after it has happened.
(a) If y is the number of people who have heard the news at a certain time, what is the formula for the number of people who have NOT heard the news:
number of people who have not heard the news =
(the only variable that should appear in your answer is y).
(b) The rate of increase in the number of people who have heard the news is proportional to the number of people who have not heard the news. Translate this into a differential equation, using k as the constant of proportionality:
y?= .
(c) A scandal in City Hall erupts. At time t=0 we assume no one has heard the news, or y(0)=0. Solve the initial value problem to determine a formula for y(t). [The formula will have the unknown constant k. Just remember that this is a constant, and you just happen to not know it.]
y(t) =
(d) 7 days after the scandal was reported, a poll showed that 450000 people have heard the news. Using this information, solve for k.
k =
(e) According to this model, about how many people will have heard the news after 12 days? Do not round your answer.
Let y(t) denote the number of people who have heard the news t days after it has happened.
(a) If y is the number of people who have heard the news at a certain time, what is the formula for the number of people who have NOT heard the news:
number of people who have not heard the news =
(the only variable that should appear in your answer is y).
(b) The rate of increase in the number of people who have heard the news is proportional to the number of people who have not heard the news. Translate this into a differential equation, using k as the constant of proportionality:
y?= .
(c) A scandal in City Hall erupts. At time t=0 we assume no one has heard the news, or y(0)=0. Solve the initial value problem to determine a formula for y(t). [The formula will have the unknown constant k. Just remember that this is a constant, and you just happen to not know it.]
y(t) =
(d) 7 days after the scandal was reported, a poll showed that 450000 people have heard the news. Using this information, solve for k.
k =
(e) According to this model, about how many people will have heard the news after 12 days? Do not round your answer.
Solution
(a) No. of people who have heard the news = y =>No. of people who haven\'t heard = 900000-y (b) Rate of increase = k *(No. of people who have not heard the news) d(y(t))/dt =k*(900000-No. of people who have heard the news) =k*(900000-y(t)) =>d(y(t))/(900000-y(t)) = k.dt =>-[ln(900000-y(t)) - ln(900000-y(0))] = k*t =>ln[(900000-y(0))/(900000-y(t)] = k*t =>{900000 - y(0)}/{900000-y(t)} = e^(k*t) =>900000-y(t) = {900000 - y(0)}*e^(-kt) y(t) = 900000- {900000 - y(0)}*e^(-kt) (c)y(0) = 0 =>y(t) = 900000- {900000}*e^(-kt) = 900000{1-e^(-kt)} (d) for t= 7 , y(t) = 450000 =>450000 = 900000{1-e^(-k*7)} =>{1-e^(-k*7)} = 0.5 =>e^(-k*7) = 0.5 =>-7k = -0.693 =>k = 0.099 (e)For t = 12 y(t) = 900000{1-e^(-0.099*12)} = 900000(1-0.305) = 625721.9 = 625722
