Question

a culture started with 6000 bacteria . after 5 hours it grew 7800 bacteria. predict how many bacteria will be present after 13 hours . use the formula P = Ae^kt round the nearest whole number

Answer 1

We know that we need to use the formula:
`P=Ae^(kt)`
where

`P` is final population of bacteria after
`t` hours

`A` is the initial population

`k` is the growth rate

`t` is the time in hours

The first thing we are going to do is find
`k`. We know for our problem that:
`P=7800`,
`A=6000`, and
`t=5`. Lets replace those values in our formula:

`P=Ae^(kt)`

`7800=6000e^(5k)`
Now, to find
`k` we are going to isolate
`e^(5k)`, and then apply logarithms:

`(7800)/(6000) =e^(5k)`

`e^(5k)= (13)/(10)`

`ln(e^(5k))=ln( (13)/(10) )`

`5k=ln( (13)/(10))`

`k= (ln( (13)/(10)) )/(5)`

`k=0.052`

Now that we have
`k`, we are going to use our formula one more time, but this time
`t=13`:

`P=Ae^(kt)`

`P=6000e^{(0.052)(13)`

`P=6000e^(0.676)`

`P=11796`

We can conclude that the culture will have 11796 bacteria after 13 hours.