A particular strain of bacteria doubles in population every 10 minutes. Assuming you start with a single bacterium in a petri dish, how many bacteria will there be in 2.5 hours?

Question

Nathan Axel · Accepted Answer

#32,768#

The trick here is to realize that you can express the increase in population as a power of #2#.

You know that every #10# minutes, the number of bacteria will double. If you take #x_0# to be the initial number of bacteria, you can say that

#x_0 * 2 -># after #10# minutes

#(x_0 * 2) * 2 = x_0 * 2^color(red)(2) -># after #color(red)(2) * 10# mintues

#(x_0 * 2^2) * 2 = x_0 * 2^color(red)(3) -># after #color(red)(3) * 10# minutes

#(x_0 * 2^3) * 2= x_0 * 2^color(red)(4) -># after #color(red)(4) * 10# minutes #vdots#

and so on. As you can see, you can say that the number of bacteria present after #t# minutes, #x#, will be

#color(purple)(|bar(ul(color(white)(a/a)color(black)(x = x_0 * 2^n)color(white)(a/a)|)))#

Here

#n# - the number of #10#-minute intervals that pass in #t# minutes

In your case, you know that #t# is equal to

#2.5 color(red)(cancel(color(black)("h"))) * "60 min"/(1color(red)(cancel(color(black)("h")))) = "150 minutes"#

So, how many #10#-minute intervals do you have here?

#n = (150 color(red)(cancel(color(black)("min"))))/(10color(red)(cancel(color(black)("min")))) = 15#

Since you start with a single bacterium in a Petri dish, you have #x_0 = 1# and

#color(green)(|bar(ul(color(white)(a/a)color(black)(x = "1 bacterium" * 2^15 = "32,768 bacteria")color(white)(a/a)|)))#

Eva Adamson · Answer

undefined To find out how many bacteria will be present in 2.5 hours, we need to determine the number of doubling periods within that time frame. Since the bacteria double in population every 10 minutes, there are 2.5 hours x 60 minutes/hour / 10 minutes/doubling period = 15 doubling periods. Therefore, the number of bacteria after 2.5 hours will be 2^(15) times the initial number of bacteria, which is 2^15 = 32,768 bacteria.