The number of bacteria in a culture grew from 275 to 1135 in three hours. How do you find the number of bacteria after 7 hours?

1 Answer
May 19, 2017

#7381#

Explanation:

Bacteria undergo asexual reproduction at an exponential rate. We model this behavior using the exponential growth function.

#color(white)(aaaaaaaaaaaaaaaaaa)color(blue)(y(t) = A_(o)*e^(kt)#

Where

  • #"y("t") = value at time ("t")"#
  • #A_("o") = "original value"#
  • #"e = Euler's number 2.718"#
  • #"k = rate of growth"#
  • #"t = time elapsed"#

You are told that a culture of bacteria grew from #color(red)[275# to #color(red)[1135# in #color(red)"3 hours"#. This should automatically tell you that:

  • #color(blue)[A_("o")# = #color(red)[275]#

  • #color(blue)["y"("t")]# = #color(red)["1135"]#, and

  • #color(blue)"t"# = #color(red)["3 hours"]#

Let's plug all this into our function.

#color(white)(aaaaaaaaaa)color(blue)(y(t) = A_(o)*e^(kt)) -> color(red)1135 = (color(red)275)*e^(k*color(red)3)#

We can work with what we have above because we know every value except for the #"rate of growth", color(blue)[k]"#, for which we will solve.

#color(white)(--)#

#ul"Solving for k"#

  • #color(red)1135 = (color(red)275)*e^(k*color(red)3)#

  • #stackrel"4.13"cancel[((1135))/((275))] = cancel[(275)/(275)]e^(k*3)#

  • #4.13 = e^(k*3)#

  • #color(white)(a)_(ln)4.13 = color(white)(a)_cancel(ln)(cancele^(k*3))#

  • #1.42 = k*3#

  • #stackrel"0.47"cancel[((1.42))/((3))] = k*cancel[(3)/(3)#

  • #0.47 = k#

Why did we figure all this out? Didn't the question ask to solve for the number of bacteria after #"time = 7 hours"# and not for #color(blue)[k], "the rate of growth"#?

The simple answer is that we needed to figure out the #"rate of growth"# so that from there we can figure out the value at time #(7)# by setting up a new function since we will have only 1 unknown left to solve.
#color(white)(--)#

#ul"Solving for number of bacteria after 7 hours"#

#color(blue)(y(t) = A_(o)*e^(kt)) -> y = (275)*e^(0.47*7)#

#y = (275)*e^(3.29)#

#y = (275)*(26.84)#

#y = 7381#

So, the bacteria colony will grow to #7381# in number after #"7 hours"#