Question

89. A scientist is studying the growth of a population of bacteria. At the beginning of her study, she has 800 bacteria. She notices that the population is quadrupling every hour.

c. Find the time, in hours, when there will be 5,120,000 bacteria. Express your answer as a logarithmic expression.

Answer 1

`(log(6400))/(log(4))`

Explanation:

The initial population of bacteria is 800 and we know that this number is quadrupling every hour.

We're going to find a function in terms of t (time) that gives us the population of bacteria at that time.

Since the population is quadrupling every hour the function in terms of t (where t is expressed in hours) is:

`f(t)=800(4^(t))`

Now we need to find the time when there will be 5,120,000 bacterias. This means the time t when f(t) = 5,120,000

So we have 5,120,000 =
`800(4^(t))`

`5,120,000 = 800(4^(t))\\(5,120,000)/(800) =4^(t) \\6400=4^(t) \\log(6400) =t log(4)\\(log(6400))/(log(4)) =t`

Therefore, the time when there will be 5,120,000 bacterias will be:

`(log(6400))/(log(4))`