From Stewart 7e pg 614 # 9
"One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard th eremor and the fraction who have not hear the rumor.
a) Write a differential equation that is satisfied by y.
b) Solve the differential equation
c) A small town has 1000 inhabitants. At 8 am 80 people have heard a rumor. By noon half the town has heard it. At what will 90 percent of the population have heard the rumor?
"
The wording of this is very ambiguous to me and I can't really make sense of it.
They mention a product, so I know that something is being multiplied and that y is a fraction which belongs to the population who have seen it so I think that "have not" heard is a constant, and that y is a fraction taht represents who have. I tried to set this up and it is the wrong answer. I am not sure what they want from that, the English usage is too ambiguous to make sense of it. The complete lack of punctuation is what really does it.
Best Answer
A start: Let $y=y(t)$ be the fraction who have heard by time $t$. Then the fraction who have not is $1-y$. The rate of change of $y$, we are told, is proportional to the product $y(1-y)$. Our differential equation is therefore $$\frac{dy}{dt}=ky(1-y).$$ This is a special case of the logistic equation, which you know how to solve.
It is convenient to let $t=0$ at $8\colon00$. So $y(0)=\frac{80}{1000}$. We are told that $y(4)=\frac{1}{2}$. These two items are enough to tell us everything about the equation, including the constant $k$. Some algebraic manipulation will be needed.
Now that you have the equation for $y(t)$ in terms of $t$, you can find the $t$ such that $y(t)=0.9$. Note that this $t$ is the time elapsed since $8\colon00$ AM. You will need to give the answer in clock terms.