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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.

User Paldepind
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1 Answer

5 votes

Answer:

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

User Minion
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