A bacterial culture (measured in millions of cells) grows according to dB dt =0.5B…

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

Problem

A bacterial culture

B(t)

(measured in millions of cells) grows according to

\frac{dB}{dt}=0.5B\,(300-B),\quad t\ge 0,

where time

t

is measured in hours. (a) If

B(0)=50

, determine the limiting value of

B(t)

t\to\infty

. (b) Using the differential equation, determine whether the population is increasing or decreasing at

t=0

when

B(0)=50

, and justify your conclusion numerically. (c) If the initial amount is

B(0)=350

, determine the limiting value of

B(t)

t\to\infty

and state whether the culture will grow or decay toward that limit. Include justification based on the sign of

\dfrac{dB}{dt}

t=0