The population of a bacterial culture grows at a rate proportional to the number of bacteria present at any time. After 3 hours it is observed that there are 400 bacteria present. After 10 hours there are 2000 present. What was the initial number of bacteria?
It is implied by context that a model for the rate of growth is
which implies that .
But the fact that , and we see that
Solving for in the first equation gives
. Substituting into the second...
and we see that
Now, again using the fact that we can conclude that
Have I done this right. If so, was there a faster way?