How do I solve this

Question

The count in a bacteria culture was 300 after 15 minutes and 1100 after 35 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?

Find the doubling period.

Find the population after 90 minutes

When will the population reach 14000.

Answer 1

C(t) = ab^t

1100 = ab^35

300 = ab^15

Divide to eliminate a

11/3 = b^20

b = 1.06712

300 = a(1.06712)^15

a = 113.2

Checking with the other equation -- VERY IMPORTANT

113.2(1.06712)^35 = 1097.8 close enough

C(t) = 113.2(1.067)^t

The initial size of the culture was 113.2

The growth rate is about 6.7% per minute

2(113.2) = 113.2(1.06712)^t

2 = 1.06712^t

t = (log2)/(log1.06712)

t = 10.7 minutes doubling time

C(90) = 113.2(1.06712)^90

C(90) = 39178

14000 = 113.2(1.06712)^t

123.675 = 1.06712^t

t = (log123.675)/(log1.06712)

t = 74.16 minutes

FYI (not part of the problem)

We could have used the Rule of 70 to estimate the doubling time.

The Rule of 70 says that the growth rate times the doubling time will be approximately equal to 70.

So in our case doubling time would be approximately 70/6.7 = 10.44 good estimate as you can see.