A biologist models the mass (in grams) of a bacterial culture with the differential…

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

Problem

A biologist models the mass

m

(in grams) of a bacterial culture with the differential equation

\frac{dm}{dt}=0.5m-0.0002m^{2},

where

t

is measured in hours. (a) Rewrite

\dfrac{dm}{dt}

in the form

k m (A-m)

, identifying the constants

k

and

A

. (b) Using your result from part (a), determine the mass value that the culture approaches as

t\to\infty

. State appropriate units. (c) Explain, in the context of the bacterial culture, the practical meaning of the value found in part (b). (d) Compute

\dfrac{dm}{dt}

when

m=\dfrac{A}{2}

and explain why this calculation is relevant in assessing the growth dynamics of the culture.