Question

For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double

Answer 1

`P(t)=256*e^(0.068185t)`

10 minutes for the population to double.

Explanation:

The general formula for the population equation, P(t), is:

`P(t)=A*e^(nt)`

We are given that for t =5 minutes and t = 2 minutes:

`P(5)=360=A*e^(5n)\\P(20)=1,000=A*e^(20n)\\\\`

Solving for A and n:

`ln(360)=ln(A)+5n\\ln(1,000)=ln(A)+20n\\ln(1,000)-4ln(360)=ln(A)-4ln(A)+20n-20n\\ln(A)=5.5455536\\A=256\\n=(ln(360)-ln(256))/(5)\\n=0.068185`

The exponential equation that represents this situation is:

`P(t)=256*e^(0.068185t)`

The population will double when P(t) = 512 bacteria:

`512=256*e^(0.068185t)\\ln(2)=0.068185t\\t=10.17\ minutes`

To the nearest minute, it takes roughly 10 minutes for the population to double.