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:
[tex]f(t)=256e^{0.06111t}[/tex]
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:
[tex]f(0)=256e^{0.06111\cdot 0}=256\cdot 1=256[/tex]
Now we find the time when the population is 5*256=1,280. The equation to solve is:
[tex]1,280=256e^{0.06111t}[/tex]
Dividing by 256:
[tex]e^{0.06111t}=1,280/256=5[/tex]
Taking natural logarithms:
[tex]ln\left(e^{0.06111t}\right)=ln5[/tex]
Applying the logarithm properties:
[tex]0.06111t=ln5[/tex]
Solving for t:
[tex]t=ln5/0.06111=26.34[/tex]
It will take 26.34 minutes for the population to reach 5 times its initial value
