A bacteria culture starts with 660 bacteria and grows at a rate proportional to its size. After 6 hours there will be 3960 bacteria. (a) Express the population after t hours as a function of t. population: (function of t) (b) What will be the population after 3 hours?

a)

b)

Step-by-step explanation:

For this case we have the following info:

represent the initial amount of bacteria

represent the population of bacteria after 6 hours

Part a

For this case we use the proportional model given by:

Where P represent the population at time t and k is a constant of proportionality.

We can rewrite the model like this:

And if we integrate both sides we got:

Now if we apply exponential we got:

For this case using the initial condition:

We have the following model:

Using the second condition we have this:

We can divide both sides by 660 and we got:

Now we can apply natural log on both sides and we got:

So then the model would be given by:

Part b

For this case we just need to replace t=3 into the model and we got:

answer: (3,11) is your answer.