Question

This exercise uses the population growth model. The count in a culture of bacteria was 600 after 2 hours and 38,400 after 6 hours. What was the initial size of the culture?

Answer 1

Let Population growth is increasing exponentially

A=A°(e^kt)

A= population at time t hours

A°=initial population

k =constant

t= time in hours

Given A at 2 hours =600,

=>600=A°(e^(k*2))

=>A°=600/(e^(2k))

And A at 6 hours=38400

=>38400=A°(e^(k*6))

=>A°=38400/(e^(6k))

Therefore we have

38400/(e^(6k))=600/(e^2k)

=>38400/600=e^(6k-2k)

=>e^4k=64

=>4k=ln(64). [ln(e^a) =a]

=>k=ln(64)/4

k=1.0397

Now,

Therefore, A°=600/e^(2*1.0397)=600/8=75

Therefore the initial size of the culture is 75.