The number of yeast cells in a laboratory culture increases rapidly initially, b
ID: 2859810 • Letter: T
Question
The number of yeast cells in a laboratory culture increases rapidly initially, but levels off eventually toward the maximum population the culture can sustain. The number of cells, (henceforth known as the Population), is modeled by the function:
n=f(t)= a/1+be^-0.7t
Where t is time measured in hours. At time t = 0, the population is 20 cells and is increasing at a rate of 12 cells/hour.
a) What does dn/dt represent and what are its units? Explain your conclusion.
b) At any value of t > 0, what would you expect the sign dn/dt of to be? Explain your conclusion.
c) Compute f ' (t) Is this derivative the same as dn/dt and does it verify your conclusion to part b? Are there any assumption that need to be made about the values of a and b?
Explanation / Answer
a) dn/dt represent the rate of change of number of cells with respect to time . Its units are cells/hour . Explanation is that when differentiate a quantity with respect to time , you get the rate at which the quantity is changing
b) if n is increasing then dn/dt >0 else if n is decreasinf dn/dt <0 . In the question it is given that cells increase rapidly and ecentually to a maximum level and maintain that level, So , dn/dt>0
c) f dash (t) = -0.7*be^(-0.7t) . This is same as dn/dt
It does verify when the coefficient is positive so for the coeffient to be positive , we need to have b as -ve