Question

Let's say that a certain antibiotic kills approximately 1/2 of the germs on the first day, 1/2 of the remaining germs on the second day, and so on.

(a) After 8 days, what fraction of the germs remain?

(b) If you had 90 million germs in your body at the beginning, how many germs will be alive at the end of the eighth day? (Round your answer to the nearest whole number.)

Answer 1

After 8 days, 1/256 of the original germs remain. If you started with 90 million germs, approximately 351,563 germs would be alive after the eighth day, when rounded to the nearest whole number.

Step-by-step explanation:

To determine what fraction of the germs remain after 8 days when an antibiotic kills approximately 1/2 of the germs each day, we can recognize this pattern as exponential decay. The fraction of germs remaining after each day can be represented as (1/2)n, where n is the number of days.

Part (a): Fraction of Germs Remaining After 8 Days

The fraction of germs remaining after 8 days is (1/2)8. Calculating this, we get:

Fraction remaining = (1/2)8 = 1/256

Part (b): Number of Germs Alive After the Eighth Day

To find out how many germs would be alive from the original 90 million germs at the end of the 8th day, we use the fraction obtained in part (a):

Number of remaining germs = 90,000,000 × (1/256)

Performing this multiplication, we get approximately 351,562.5, which we round to the nearest whole number, yielding 351,563 germs.