Question

Suppose Y(t) = 25e^3t + 12 represents the number of bacteria present at time t minutes. At what time will the population reach 100 bacteria? ( Note: Answers are expressed in terms of natural logarithm

Answer 1

Given:

The given function is:

$Y(t)=25e^{3t}+12$

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,

$Y(t)=25e^{3t}+12$

Putting $Y(t)=100$, we get

$100=25e^{3t}+12$

$100-12=25e^{3t}$

$88=25e^{3t}$

Divide both sides by 25.

$\dfrac{88}{25}=e^{3t}$

Taking ln on both sides, we get

$\ln (\dfrac{88}{25})=\ln e^{3t}$

$\ln(\dfrac{88}{25})=3t$                  $[\because \ln e^x=x]$

Divide both sides by 3.

$\dfrac{1}{3}\ln(\dfrac{88}{25})=t$

Therefore, the required time is $\dfrac{1}{3}\ln(\dfrac{88}{25})$ minutes.