A culture of bacteria grows at a rate proportional to the number of bacteria present. If 40,000 bacteria are present at the start of an experiment, and if the number of bacteria doubles in 3 hours, how many bacteria will be present after 5 hours? (
Answers
Given : A culture of bacteria grows at a rate proportional to the number of bacteria present . 40,000 bacteria are present at the start of an experiment, number of bacteria doubles in 3 hours,
To find : how many bacteria will be present after 5 hours
Solution:
Let N be the number of bacteria present at time t
bacteria grows at a rate proportional to the number of bacteria present.
dN/dt = kN
=> dN/N = k dt
integrating both sides
=> ln (N) = kt²/2 + c
at t = 0 Bacteria = 40000
=> ln (40000) = c
=> c= 10.6
ln (N) = kt²/2 + ln (40000)
number of bacteria doubles in 3 hours
=> N = 80000 at t = 3
=> ln(80000) = k(9/2) + ln(40000)
=> 0.693 = k(9/2)
=> k = 0.154
ln (N) = 0.154 t²/2 + ln (40000)
=> t = 5 hrs
=> ln (N) = 0.154 (5²/2) + ln (40000)
=> ln(N/40000) = 1.925
=> N/40000 = 6.855
=> N = 2,74,200
2,74,200 Bacteria Present after 5 hours
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