Problema Solution
during a research experiment it was found that the number of bacteria in a culture grew at a rate proportional to its size. at 5am there were 3000bacteria present in the culture. at noon, the number of bacteria grew to 4100. how many bacteria will there be by midnight?
Answer provided by our tutors
Let y be the number of bacterias and t be the time.
The initial value is A = 3,000.
y = Ae^(kt)
y = 3000e^(kt), where k is a constant
Second condition says y = 4,100 when t = 7 hr (since 12 - 5 = 7).
4100 = 3000e^(7k))
k = ln(4.1/3) / 7
Finally the equation for the number of bacteria is:
y = 3000e^[ t( ln(4.1/3)/7) ]
At midnight t = 19 hr (the tome from 5 am till midnight) so we have:
y = 3000e^[ 19( ln(4.1/3)/7)]
y = 3000 e^(ln (4.1/3)^(19/7))
y = 3000* (4.1/3)^(19/7)
y = 7,004 bacteria.
y = 2,544