The count in a bacteria culture was 500 after 10 minutes and 1300 after 40 minutes.

Question

The count in a bacteria culture was 500 after 10 minutes and 1300 after 40 minutes. Assuming the count grows exponentially,

1. What was the initial size of the culture?   
2. Find the doubling period.   
3. Find the population after 95 minutes.   
4. When will the population reach 13000. 

Answer 1

500= Pe^10r

1300 =Pe^40r

where P=initial amount and r=rate of change in the bacteria

divide 2nd equation by 1st

13/5 =

2.6 = e^30r

take natural logs of both sides

30r = ln2.6

r = (ln2.6)/30= about .0319 = 3.19% per minute

P=500/e^10r = 500/e^.319

= about 363.43 = initial amount of bacteria

doubling period > 10 minutes, but < 40 minutes

2 = e^.0319t

2 = e^.0319t

.0319t = ln2

t = (ln2)/.0319=about 21.729 minutes

after 95 minutes

A = 363.43e^.0319(95) =about 7,525.76

it reaches 13000 when

13000= 363.43e^.0319t

e^.0319t = 13000/363.43

.0319t = ln(13000/363.43)

t = ln(13000/363.43)/.0319 = about 112.135 minutes