We are told that the bacteria is growing at a "rate proportional to the number of bacteria present at any time", thus;
for some (Since we are told that the rate of bacteria is growing)
Now solve the differential using the change of variable method.
where is the constant retained from both integrals.
Now rearrange for .
Firstly we are told that "after 3 hours...there are 400 Bacteria present", hence subbing into we get;
Secondly we are told that "After 10 hours there are 2000 bacteria present", thus subbing into we get;
Now we have to solve equations and simultaneously. Dividing equation by equation we obtain;
Now we can work by subbing into either of the 2 equations. Note that is the number of bacteria present initially since if then