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?

1 Answer
Aug 3, 2016

Answer:

#32,768#

Explanation:

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)|)))#