A microbiology lab studies a yeast culture whose size (in millions of cells)…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

A microbiology lab studies a yeast culture whose size

Y

(in millions of cells) satisfies a logistic differential equation of the form

\frac{dY}{dt}= kY(A-Y)

. The researchers record the following two data points: • When

Y = 2.0

, the instantaneous growth rate is

\frac{dY}{dt}= 1.6

(in millions of cells per hour). • When

Y = 6.0

, the instantaneous growth rate is

\frac{dY}{dt}= 0.8

(in millions of cells per hour). Assume

k

and

A

are positive constants. (a) Set up a system of equations based on the two data points that can be used to determine

k

and

A

. (b) Solve the system to obtain exact numerical values of

k

and

A

. (c) Interpret the value of

A

in the context of the yeast culture. (d) Using your determined parameters, compute the instantaneous growth rate when

Y = 3.0

and discuss whether this rate is larger or smaller than the rate when

Y = 2.0

. Provide a brief explanation for the observed change.