The count in a bateria culture was initially 800, and after 40 minutes the population had increased to 1900.

Question

Find the doubling period. 

Find the population after 105 minutes

When will the population reach 10000?

Answer 1

This is an exponential growth type of problem where the model is:

N = N₀ x g^t where N = the number of bacteria at time t in minutes and N₀ = the number of bacterial at t = 0

So we have N = 800 x g^t and we must first figure out the value if g. We know

1900 = 800 x g⁴⁰ or 1900 / 800 = 2.375 = g^t now take the log of both sides and use the property of logs

log ( 2.375 ) = 40 x log ( g ) and log ( g ) = log ( 2.375 ) / 40 = .00939159 So b we know is

b = 10^.00939159 = 1.0218604 So uur exponential growth function is

N = 800 x 1.0218604^t Now we can answer the questions.

I) Find the doubling period. We have

1600 = 800 x 1.0218604^t or 1600 / 800 = 2 = 1.0218604^t Taking the logs of both sides

t = log ( 2 ) / log ( 1.0218604 ) = 32.04 minutes

II) Find the population after 105 minutes

N = 800 x 1.0218604¹⁰⁵ = 7748.27

III) When will the population reach 10,000? We have

10,000 = 800 x 1.0218604^t So

10,000 / 800 = 12.5 = 1.0218604^t

t = log ( 12.5 ) / log ( 1.0218606 ) = 116.8 minutes