Can someone please help me with this math problem?

Question

The count in a bacteria culture was 400 after 15 minutes and 1800 after 30 minutes. Assuming the count grows exponentially. You may enter the exact value or round to 2 decimal places.

1. What was the initial size of the culture?
2. Find the doubling period.
3. Find the population after 60 minutes
4. When will the population reach 15000

Answer 1

400=Pe^15r

r =[ln400/P]/15

1800=Pe^30r

r=[ln1800/P]/30 =[ln400/P]/15

2ln400-2lnP= ln1800-lnP

lnP=2ln400-ln1800= about 4.49

P = e^4.49= about 89.12 = initial amount

r = [ln400/89.12]/15 = about .1 = 10% per minute

doubling time

2=e^rt = e^.1t

ln2 = .1t

t =10ln2 = 6.931 minutes

A=89.12e^.1(60) = 89.12e^6 = about 35,953.57 in an hour = 60 minutes

15000=89.12e^.1t

t = 10[ln15000/89.12]= 51.26 minutes to reach 15000

rough calculations to see if the numbers are reasonable

about 90 at t=0

doubles in about 7 minutes

180 t = 7

360 14

400 15 looks about right

800 22

1600 29

1800 30 in the right range

3600 37

7200 45

14400 52

15000 at 51 minutes is in the right range

30,000 58

35,000 in 60 minutes is about right