Question

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

Answer 1

Answer:

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

Step-by-step 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