Question

The number of bacteria in a culture grew from 275 to 1135 in three hours. Find the amount of time needed for the number of bacteria to grow to 5000.

Answer 1

6.1 hours

Explanation:

Given : The number of bacteria in a culture grew from 275 to 1135 in three hours.

To find : Amount of time needed for the number of bacteria to grow to 5000.

Solution : Let the bacteria undergo through an exponential growth

`y(t)=A e^(kt)` where,

y(t)= value at time t

A= original value

t=time

k = rate of growth

First, we find the rate of growth in culture grew from 275 to 1135 in three hours.

`y(t)=A e^(kt)`

`1135=275 e^(3k)`

`(1135)/(275)=e^(3k)`

`4.13=e^(3k)`

Taking ln(natural log) both side

`ln(4.13)=3k`

`1.42=3k`

`k=0.47`

Now, we find the time needed for the number of bacteria to grow to 5000

`y(t)=275 e^(0.47t)`

`5000=275 e^(0.47t)`

`(5000)/(275)=e^(0.47t)`

`18.18=e^(0.47t)`

Taking ln(natural log) both side

`ln(18.18)=0.47t`

`2.900=0.47t`

`(2.900)/(0.47)=t`

`t=6.17`

Therefore, Amount of time is approx 6.1 hours

Answer 2

6.1 hours

Explanation:

The starting number 275 is multiplied by the factor 1135/275 in 3 hours. This gives rise to the exponential model ...

... n(t) = 275·(1135/275)^(t/3)

We want to find t when n(t) = 5000. Substituting this into the equation and solving, we have ...

... 5000 = 275·(1135/275)^(t/3)

... 5000/275 = (1135/275)^(t/3) . . . . divide by 275

... log(5000/275) = (t/3)·log(1135/275) . . . . take the log

... t = 3·log(5000/275)/log(1135/275) . . . . multiply by the inverse of the x-coefficient

... t ≈ 6.13795 . . . hours

Rounded to reasonable precision, the time is approximately 6.1 hours.