It takes 28 minutes for a certain bacteria population to double. If there are 4,241,763 bacteria in this population at 1:00 p.m., which of the following is closest to the number of bacteria in millions at 4:30 pm on the same day?
time from 1:00 to 4:30 is 3.5 hours
28 minutes = 28/60 or .466667 hours
N = 4241763(2)^(3.5/.466667)
= 4241763(2)^7.5
= appr. 767 841 122
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To solve this problem, we need to determine the number of times the population doubles from 1:00 p.m. to 4:30 p.m. Let's calculate the time difference between these two points:
4:30 pm - 1:00 pm = 3 hours and 30 minutes = 3.5 hours
Since the bacteria population doubles every 28 minutes, the number of times it doubles in 3.5 hours can be found by dividing 3.5 by 28:
3.5 hours / 28 minutes = 0.125 times
Therefore, the bacteria population will double approximately 0.125 times from 1:00 pm to 4:30 pm.
Now, let's calculate the final population by multiplying the initial population by 2 raised to the power of the number of times it doubles:
Final population = (Initial population) * 2^(Number of times it doubles)
Final population = 4,241,763 * 2^(0.125)
Calculating this value:
Final population ≈ 4,241,763 * 2^(0.125)
≈ 4,241,763 * 1.059
Now, let's calculate the final population in millions:
Final population in millions ≈ (4,241,763 * 1.059) / 1,000,000
≈ 4.499 million
Therefore, the closest number of bacteria in millions at 4:30 pm on the same day is approximately 4.499 million.
To find the number of bacteria at 4:30 pm, we need to calculate the number of doubling periods between 1:00 pm and 4:30 pm.
First, let's determine the time difference between 1:00 pm and 4:30 pm:
4:30 pm - 1:00 pm = 3 hours and 30 minutes = 3 × 60 + 30 = 210 minutes.
Next, let's calculate the number of doubling periods within that time:
210 minutes / 28 minutes per doubling period = 7.5 doubling periods.
Since you cannot have half of a doubling period, we will consider it as 7 full doubling periods.
Now, let's calculate the number of bacteria at 4:30 pm based on these doubling periods:
Number of bacteria = Initial number of bacteria × 2^n,
where n is the number of doubling periods.
Starting with 4,241,763 bacteria at 1:00 pm, we can calculate:
Number of bacteria at 4:30 pm = 4,241,763 × 2^7 = 4,241,763 × 128 = 543,531,424.
Therefore, the closest number of bacteria in millions at 4:30 pm is 543.5 million (rounded to one decimal place).