Question

A biologist observed a population of bacteria that grew at a rate expressed by the exponential equation

f(t) = 256e0.06111)
where t is in minutes. How long will it take the population to reach 5 times its initial value?
Round to two decimal places and do not include "t =" in your answer.

Answer 1

It will take 26.34 minutes for the population to reach 5 times its initial value

Explanation:

Exponential Growing

The population of bacteria grows at a rate expressed by the equation:

`f(t)=256e^(0.06111t)`

Where t is in minutes.

We need to know when the population will reach 5 times its initial value. The initial value can be determined by setting t=0:

`f(0)=256e^(0.06111\cdot 0)=256\cdot 1=256`

Now we find the time when the population is 5*256=1,280. The equation to solve is:

`1,280=256e^(0.06111t)`

Dividing by 256:

`e^(0.06111t)=1,280/256=5`

Taking natural logarithms:

`ln\left(e^(0.06111t)\right)=ln5`

Applying the logarithm properties:

`0.06111t=ln5`

Solving for t:

`t=ln5/0.06111=26.34`

It will take 26.34 minutes for the population to reach 5 times its initial value