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.

`(88)/(25)=e^(3t)`

Taking ln on both sides, we get

`\ln ((88)/(25))=\ln e^(3t)`

`\ln((88)/(25))=3t`
`[\because \ln e^x=x]`

Divide both sides by 3.

`(1)/(3)\ln((88)/(25))=t`

Therefore, the required time is
`(1)/(3)\ln((88)/(25))` minutes.