You ingest a certain dangerous bacteria at dinner by accident. When more than 1,000,000 of these organisms are in your gut, you will become violently ill. Their population doubles every 20 minutes. You ate dinner at 7pm and by 5am you are sick. At least how many of these bacteria did you ingest at dinner?
To find out how many of the bacteria you ingested at dinner, we need to work backward from the time you became sick at 5 am.
From 7 pm to 5 am, there is a total of 10 hours, which is equal to 600 minutes. Knowing that the bacteria double every 20 minutes, we can calculate how many cycles of doubling occurred within that time:
600 minutes / 20 minutes = 30 cycles
During each cycle, the population of bacteria doubles. Therefore, we need to find the initial number of bacteria that, when doubled 30 times, would reach at least 1,000,000 organisms.
We can use the formula for exponential growth: P = P0 * 2^N, where P is the final population, P0 is the initial population, and N is the number of cycles.
Rearranging the formula, we have P0 = P / 2^N.
Let's plug in the values: P = 1,000,000 and N = 30.
P0 = 1,000,000 / 2^30 ≈ 0.93
So, at least 0.93 or approximately 1 bacterium must have been ingested at dinner for the population to double 30 times and reach 1,000,000 organisms by 5 am when you became sick.
N = a (2)^(t/20), where N is the number of bacteria, t is the number of minutes, a is the initial count, our problem.
counting 7:00 as zero time
and 5:00 am as t = 10 hrs = 120 minutes
1000000 = a(2)^(120/20)
1000000 = a(2)^6
64a = 1000000
a = 1000000/64 = 15625 of the germs
you could work backwards and cut the count in half 6 times
1,000,000 ---> 500,000
500,000 ---> 250,000
250,000----> 125,000
125,000----> 62,500
62,500 -----> 31250
31250 ---> 15625