Question

A research student is working with a culture of bacteria that doubles in size every

five minutes. The initial population count was 1000 bacteria. Write an exponential
equation representing this situation. What is the population size after 2 hours?

Answer 1

`2^(24)*1000` bacteria

Explanation:

The generic form of an exponential function is:
`f(n)=ar^n` , where a is the initial amount, r is the rate, and n is the time.

In this case, the initial amount is 1000, so a = 1000. Also, the rate is 2 because the culture doubles in size every 5 minutes. However, our n is not just n; it's actually n/5 because the rate is doubling per 5 minutes, not per 1 minute. If it was per 1 minute, then the time would simply be n. Unfortunately, that's not the case here, so the exponent of r should be n/5, where n is the number of minutes that have passed.

Now, we have:
`f(n)=1000(2)^(n/5)`

Since n is the number of minutes, we need to convert 2 hours to minutes:

2 hours * 60 min/hour = 120 minutes.

`f(120)=1000(2)^(120/5)=1000(2)^(24)=2^(24)*1000`
`f(120)=1000(2)^(120/5)=1000(2)^(24)=2^(24)*1000`

Thus, the answer is
`2^(24)*1000` bacteria.

Hope this helps!

Answer 2

Exponential formula:

Total = start value x 2 ^ time/time it takes to double:

Total = 1000 x 2 ^ 120/5

Total = 1000 x 2^24 Bacteria

Total = 1.67 x 10^10 bacteria