Given that a bacteria whose growth rate is proportional to the amount of bacteria present find how much the population will increase at the end of two hours if it doubles after one hour.

We are given that the population doubles every hour. Therefore, the population will be four times larger than the starting population after two hours.

*Related*

Your reasoning is incomplete, it doesn’t necessarily double each hour. y’ = ky since it’s rate of growth is proportional to the amount present, so y=f(t)=ce^kt (this is an exercise in Apostol’s Calculus) and since f(1)=2f(0)=2c we have e^k=2 so k=log2 and thus f(2)=ce^2log2=4c which coincidentally matches your solution in fact f(t)=c2^t

I agree