Given:
The given function is:
[tex]Y(t)=25e^{3t}+12[/tex]
Where Y represents the number of bacteria present at time t minutes.
To find:
The time taken by bacteria population to reach 100 bacteria.
Solution:
We have,
[tex]Y(t)=25e^{3t}+12[/tex]
Putting [tex]Y(t)=100[/tex], we get
[tex]100=25e^{3t}+12[/tex]
[tex]100-12=25e^{3t}[/tex]
[tex]88=25e^{3t}[/tex]
Divide both sides by 25.
[tex]\dfrac{88}{25}=e^{3t}[/tex]
Taking ln on both sides, we get
[tex]\ln (\dfrac{88}{25})=\ln e^{3t}[/tex]
[tex]\ln(\dfrac{88}{25})=3t[/tex] [tex][\because \ln e^x=x][/tex]
Divide both sides by 3.
[tex]\dfrac{1}{3}\ln(\dfrac{88}{25})=t[/tex]
Therefore, the required time is [tex]\dfrac{1}{3}\ln(\dfrac{88}{25})[/tex] minutes.
