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 the rumor and the fraction who have not heard the rumor. (a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.) dy dt = Correct: Your answer is correct. (b) Solve the differential equation. (Let y(0) = y0.) (c) A small town has 4500 inhabitants. At 8 AM, 360 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Do not round k in your calculation. Round your final answer to one decimal place.) hours after 8 AM

a) The differential equation for this model is:

b) The model y(t) is:

c) t=7.6 hours after 8 AM

Step-by-step explanation:

The rate of spread can be expressed as the first derivative dy/dt, being y the fraction of the population that have heard the rumor.

We know that is proportional to the fraction of the population that have heard the rumor (y) and to the fraction of the population that have not already heard the rumor (y-1). The constant of porportionality is k.

The differential equation becomes then:

Solving this differential equation:

The initial condition is y(0)=y0

Then the model for y(t) becomes:

c) The initial proportion that have heard the rumor is

Then we have:

We also know that at noon (t=4 hours after 8 AM), the fraction y(4) is 0.5.

Then, we can calculate k from there:

Now we can calculate t, so that y(t)=0.90.