Suppose a bacteria culture grows at a rate that is proportional to the populatio

Question

Suppose a bacteria culture grows at a rate that is proportional to the population present at the time t. If the bacteria population (in mu g) triples in 2 days and the initial population is 0. 2 mu g, what will the population be in 1 week? Write a differential equation which models the number y of people with the innovation after t days. Make sure to include the initial conditions Rewrite the differential equation as a Bernoulli differential equation. Solve it as a Bernoulli differential equation using the correct substitution.

Explanation / Answer

Let "N" be the nummber of bacteria present at any given time "t".
So the rate of change of bacteria population will be dN/dt

(a) dN/dt ? N
dN/dt = kN
dN/N = (dt)k

If we integrate this, we get
lnN = kt + c
N = e*(kt + c) [e to the power of (kt + c)]


(b) Let's use the euqation we got above: lnN = kt + c
ln(5 x 10*6) = c (since t = 0 so kt = 0)....... equation 1
ln(7 x 10*6) = 3k + c ........... equation 2

so we substitute the value of c from eq 1 into eq 2 to get
ln(7 x 10*6) = 3k + ln(5 x 10*6)
3k = 0.3365
k = 0.112 approximately

and c = 15.425

so the equation that expresses the number of millions of bacteria as a function of the number of hours will be:
N = e*(0.112t + 15.425)

(c) You'll have to first write up a table of Number of bacteria (N) and time in hours (t). Then put in differnt values of "t" in the above equation ( t = 0, 1, 2 ,3 4 etc) and find the corresponding values of "N". Once you have filled the table with 5 or 6 values for each N and t, then draw a graph using those values. It will be a curve.

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