Question

A biologist recorded a count of 348 bacteria present in a culture after 5 minutes and 1088 bacteria present after 20 minutes. To the nearest whole number, what was the initial population ?

Answer 1

The initial bacteria population is 238

Explanation:

Let us have the increase as an exponential equation

P(t) = I(1 + r)^t

where P(t) is the count at a particular time

r is the percentage increase

I is the initial population

t is the time

So for 5 minutes, we have

348 = I(1 + r)^5

For 20 minutes

1088 = I(1 + r)^20

Divide the second equation by the first

1088/348 = I(1 + r)^20/I(1 + r)^5

3.13 = (1 + r)^15

Take the ln of both sides

ln 3.13 = 15 ln (1 + r)

ln 3.13/15 = ln (1 + r)

0.0761 = ln (1 + r)

1 + r = e^0.0761

1 + r = 1.079

r = 1.079-1

r = 0.079 which is 7.9%

So per minute. we have an increase of 7.9%

So let’s get I by substituting

From;

348 = I(1 + 0.079)^5

348/1.079^5 = I

I = 237.9 which is 238 to the nearest whole number