We answer the first part of the problem here.

`P = 50000`, so our differential equation becomes:

`(dA)/(dt)=50\ 000cA-cA^2`

`=cA(50\ 000-A)`

Solving using Scientific Notebook gives us the number of people affected at time t:

`A(t)=(50\ 000)/(1+499e^(-50\ 000ct))`

[We'll see how to solve this kind of problem in the Separation of Variables page, later in this chapter.]

Now, we are told that at `t = 10`, `A = 1000`.

So now we can substitute and solve for `c`:

`c = 4. 6416 xx 10^-6`

So substituting this value of c into the expression for A(t) gives us:

`A(t)=(50\ 000)/(1+499e^(-0.23208t))`

Easy to understand math videos: