7. A certain strain of bacteria doubles every 6 minutes. Assuming that you start with only one bacterium, how many bacteria could be present at the end of 84 minutes (round answer to the nearest bacteria)?
2^(84/6) = 2^14 = 16384
To find the number of bacteria at the end of 84 minutes, we need to determine how many times the bacteria double during this period.
Since the bacteria double every 6 minutes, we can divide the total time (84 minutes) by the doubling time (6 minutes):
84 minutes / 6 minutes = 14
This means that the bacteria double 14 times during the 84-minute period.
Starting with one bacterium, we can calculate the number of bacteria at the end by raising 2 to the power of the number of doublings:
2^14 = 16,384
Therefore, there could be approximately 16,384 bacteria present at the end of 84 minutes.
To solve this problem, we need to calculate the number of bacteria at each interval of 6 minutes and find the final count at the end of 84 minutes.
The bacteria double every 6 minutes, which means the growth rate is 2 (doubling every interval). So, we can use the formula:
Population = Initial Population * Growth Rate^(Time / Interval)
In this case, the initial population is 1 (starting with one bacterium), the growth rate is 2, and the time interval is 6 minutes.
Let's calculate the population at each interval:
At 6 minutes: Population = 1 * 2^(6/6) = 1 * 2^1 = 1 * 2 = 2
At 12 minutes: Population = 1 * 2^(12/6) = 1 * 2^2 = 1 * 4 = 4
At 18 minutes: Population = 1 * 2^(18/6) = 1 * 2^3 = 1 * 8 = 8
At 24 minutes: Population = 1 * 2^(24/6) = 1 * 2^4 = 1 * 16 = 16
At 30 minutes: Population = 1 * 2^(30/6) = 1 * 2^5 = 1 * 32 = 32
At 36 minutes: Population = 1 * 2^(36/6) = 1 * 2^6 = 1 * 64 = 64
At 42 minutes: Population = 1 * 2^(42/6) = 1 * 2^7 = 1 * 128 = 128
At 48 minutes: Population = 1 * 2^(48/6) = 1 * 2^8 = 1 * 256 = 256
At 54 minutes: Population = 1 * 2^(54/6) = 1 * 2^9 = 1 * 512 = 512
At 60 minutes: Population = 1 * 2^(60/6) = 1 * 2^10 = 1 * 1024 = 1024
At 66 minutes: Population = 1 * 2^(66/6) = 1 * 2^11 = 1 * 2048 = 2048
At 72 minutes: Population = 1 * 2^(72/6) = 1 * 2^12 = 1 * 4096 = 4096
At 78 minutes: Population = 1 * 2^(78/6) = 1 * 2^13 = 1 * 8192 = 8192
At 84 minutes: Population = 1 * 2^(84/6) = 1 * 2^14 = 1 * 16384 = 16384 (rounded)
Therefore, at the end of 84 minutes, there could be approximately 16,384 bacteria present.