Population growth the number of a certain type of bacteria increases continuously at a rate proportional to the number present. after 2 hours, there are 200 bacteria, and after 4 hours, there are 300 bacteria.
(a) find an exponential model given the population p after t hours.
(b) how many becteria will there be after 7 hours?
(c) how long will it take for the population to double?

a)

b)

c) For this case we need to satisfy this equation:

If we divide both sides by we got:

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

And the

Step-by-step explanation:

For this case the proportional model is given by the following differential equation:

Where P is the population, t the time and k a proportional constant.

IWe can rewrite the differential equation like this:

And if we integrate both sides we got:

Part a

And if we apply exponentiation on both sides we got:

So then our model is given by:

Where is the initial population for t =0.

For this case we have some conditions given:

So then we have two equations:

  (1)

  (2)

From equation (1) we can solve for like this:

  (3)

And if we replace equation (3) into equation (2) we got:

And we can divide both sides by 200 and we got:

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

And then the value for k is:

And since we have the value for k we can find the value or like this:

And with that our model is given by:

Part b

For this case we just need to replace t=7 and see what we got:

Part c

For this case we need to satisfy this equation:

If we divide both sides by we got:

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

And the