Question

A bacteria culture grows with constant relative growth rate. After 2 hours there are 400 bacteria and after 8 hours the count is 50,000. (a) Find the initial population. P(0) = 80 Correct: Your answer is correct. bacteria (b) Find an expression for the population after t hours.

Answer 1

The expression for the population after t hours is

`P(t) = 80e^(0.804t)`

Explanation:

According to the information given from the problem we know that

P(2) = 400

P(8) = 50 000

In general a model like this grows exponentially so the model would look like this.

`P(t) = C_0e^(kt)`

What we don't know is
`C_0,k`, and we get it from the information given. So

We know that

`400 = C_0 e^(2k)\\50000 = C_0 e^(8k)`

That's a system of equations with two variables and two unknowns, when you solve it you get that

C = 80

k = log(5)/2 = 0.804

Therefore

`P(t) = 80e^(0.804t)`