Question

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.

Answer 1

(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.