Bacteria culture with growth proportional to size

[QuantumTheory]

My friend had to do this problem in Calculus BC. I'm no good at calculus, but I decided it was be fun for someone to figure it out for me.

A bacteria culture starts with 500 bacteria and grows at a rate proportional to its size. After 3 hours there are 8000 bacteria.
(a) Find an expression for the number of bacteria after t hours.
(b) Find the number of bacteria after 4 hours.
(c) Find the rate of growth after 4 hours.
(d) When will the population reach 30,000?

I think it involves differential equations.

 

[Corneo]

but I decided it was be fun for someone to figure it out for me.

Start with $$y=Ae^{kt}$$

 

[Jayboy]

Yeah that amused me too .I'd probably start a little earlier:

Growth proportional to size implies...

$$\frac{dy}{dt}=ky$$

with the intial condition $$y(0)=500$$ and the time t = 3 (in hours!) condition $$y(3)=8000$$.

Your mission, should you choose to accept it, is to find the general solution to the DE above, and the values of k and the constant of integration A.

That's the model, the rest should be ok?