A bacterial culture starts with 800 bacteria and doubles in size every half hour.
(a)
How many bacteria are there after 5 hours?
(b)
How many bacteria are there after t hours?
(c)
How many bacteria are there after 40 minutes?
(d)
Estimate the time (in hours) for the population to reach 40,000.
(a) After 5 hours, the number of doublings is 5 * 2 = <<5*2=10>>10 doublings.
So the number of bacteria is 800 * (2^10) = 800 * 1024 = <<800*1024=819200>>819,200 bacteria.
(b) After t hours, the number of doublings is t * 2 = 2t doublings.
So the number of bacteria is 800 * (2^(2t)) = 800 * (4^t) bacteria.
(c) After 40 minutes, there are 40/30 = 4/3 hours.
So the number of doublings is (4/3) * 2 = 8/3 doublings.
Therefore, the number of bacteria is 800 * (2^(8/3)) = 800 * 8 = 6400 bacteria.
(d) Let N be the number of doublings needed to reach 40,000 bacteria.
Then 2^N = 40,000 / 800 = 50.
Taking the log base 2 of both sides, N = log2(50) ≈ 5.643856.
So it would take approximately 5.643856 hours for the population to reach 40,000 bacteria.
(a) To find the number of bacteria after 5 hours, we need to determine the number of doubling periods within that time.
Since the bacteria double in size every half hour, there are 5 hours / 0.5 hours = 10 doubling periods.
Starting with 800 bacteria, after 10 doubling periods, the total number of bacteria is 800 * (2^10) = 800 * 1024 = 819,200 bacteria.
Therefore, there will be 819,200 bacteria after 5 hours.
(b) To find the number of bacteria after t hours, we need to determine the number of doubling periods within that time.
Since the bacteria double in size every half hour, there are t hours / 0.5 hours = 2t doubling periods.
Starting with 800 bacteria, after 2t doubling periods, the total number of bacteria is 800 * (2^(2t)) bacteria.
(c) To find the number of bacteria after 40 minutes, we need to determine the number of doubling periods within that time.
Since the bacteria double in size every half hour, there are 40 minutes / 30 minutes = 4/3 doubling periods.
Starting with 800 bacteria, after 4/3 doubling periods, the total number of bacteria is 800 * (2^(4/3)) bacteria.
(d) To estimate the time for the population to reach 40,000 bacteria, we can set up the equation:
40,000 = 800 * (2^(t/0.5))
Dividing both sides of the equation by 800:
50 = 2^(t/0.5)
Taking the logarithm base 2 of both sides:
log2(50) = t/0.5
Simplifying:
log2(50) * 0.5 = t
Using a calculator, we can approximate the value of log2(50) ≈ 5.64385618977.
Therefore, t ≈ 5.64385618977 * 0.5 = 2.821928095 seconds.
So, it would take approximately 2.822 hours (or 2 hours and 49 minutes) for the population to reach 40,000 bacteria.