A certain kind of bacteria doubles in quantity every hour. At midnight two nights ago. only two of these bacteria were in a jar. At midnight last night, the jar was full. At what time was the jar half full?
It was half full at 11 p.m. last night.
Could someone explain this answer.
To determine at what time the jar was half full, we need to work backwards from the time it was full.
Let's break down the problem and calculate the intervals:
- The bacteria double in quantity every hour. This means that by the time the jar was full, it went through a series of doubling intervals.
- Given that at midnight two nights ago, there were only two bacteria in the jar, and by midnight last night, there were enough to fill the jar, we can deduce that there were 24 hours in between.
- During these 24 hours, the bacteria population doubled a certain number of times.
Now, let's calculate the number of doubling intervals:
1. Start with the number of bacteria in the jar at midnight two nights ago: 2
2. By midnight last night, the jar is full, so we can use the given information to solve for the number of doubling intervals:
Number of doubling intervals = log2(Number of bacteria at midnight last night)
= log2(Number of bacteria at midnight two nights ago * (2^(Number of hours between)))
= log2(2 * (2^24))
= log2(2^(25))
= 25 doubling intervals
3. Since during each doubling interval the population doubles, we can work backward to find the time when the jar was half full.
For each doubling interval, the population halves. Therefore, we need to go back 25 intervals from midnight last night.
Let's calculate the time when the jar was half full:
- There are 60 minutes in an hour, so 25 intervals are equal to 25 hours.
- Since there are 24 hours in a day, subtracting 25 hours gives us -1 hour.
- Therefore, the jar was half full at 11 PM two nights ago.
So, the jar was half full at 11 PM two nights ago.