Problema Solution
during a research experiment, it was found that the number of bacteria grew at a rate proportional to its size. At 5:00 AM there were 2,000 bacteria present in the culture. At noon, the number of bacteria grew to 2,700. How many bacteria will there be at midnight?
Answer provided by our tutors
A model that describes the growth of a population P, after a certain time t, is
P(t)= P*e^(kt)
where P = 400 bacteria is the initial population, t = the number of hours after 5:00 AM
At noon t = 7 and P(7) = 5100
400*e^(7k) = 5100 divide both sides by 4000
e^(7k) = 5100/4000
ln (e^(7k)) = ln (5100/4000)
7k = ln (5100/4000) divide both sides by 7
k = (1/7) ln (1.275)
by solving we find
k = 0.0347
click here to see the step by step solution of the equation:
P(t)= 4000*e^(0.0347t)
at midnight t = 24 - 5 = 19
P(19) = 4000*e^(0.0347*19)
P(19) = 7734 bacteria approximately
there will be 7734 bacteria approximately at midnight.