A bacterial culture in a laboratory Petri dish grows according to a differential…

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

Problem

A bacterial culture in a laboratory Petri dish grows according to a differential equation of the form

\dfrac{dP}{dt}=kP(M-P)

, where

P(t)

is the number of bacteria (in thousands)

t

hours after the start of the experiment. The carrying capacity is estimated to be

M=1.0

(thousand bacteria), and the proportionality constant is

k=0.20\,\text{hr}^{-1}

. Initially, there are

P(0)=0.10

thousand bacteria. (a) Write an explicit expression for

\dfrac{dP}{dt}

that incorporates the given numerical values. (b) Calculate, to three decimal places, the instantaneous growth rate in bacteria per hour at

t=0

. (c) Calculate the instantaneous growth rate when the population reaches

P=0.50

thousand bacteria, and compare this value to your result in part (b).