The count in a bacteria culture was 200 after 15 minutes and 1200 after 35 minutes. Assuming the count grows exponentially,

Question

What was the initial size of the culture?   

Find the doubling period.   

Find the population after 80 minutes.   

When will the population reach 13000

Answer 1

200=Pe^15r

r=(ln(200/P)/15

1200=Pe^35r

r=(ln(1200/P)/35

ln(1200/P)/ln(200/P)= 35/15=7/3

(ln1200- lnP)/(ln200-lnP) =7/3

3ln1200 -3lnP = 7ln200 -7lnP

4lnP = 7ln200-3ln1200

lnP= (7ln200-3ln1200)/4= about 3.9545

P= e^3.9545= 52.1696= initial amount

2P=Pe^rt

r= ln(200/P)/15= [ln(200/52.1696)]/15=.0896

2=e^.0896t

.0896t=ln2

t=ln2/.0896=7.736 minutes = doubling time

A=52.1696e^[80(.0896)]=67,676.884 after 80 minutes

13000=52.1696e^.0896t

t=[ln13000/52.1696]/.0896=61.5871 minutes to reach 13,000

a very rough check

doubling time about 8 minutes

some very rough approximations

about 50 at 0 minutes

100 about 8

200 at 15 minutes

400 23

800 31

1200 35

1600 39

3200 47

6400 53

12800 61

25600 69

50,000 77

100,000 85

maybe 68,000 at 80 minutes