A population satisfies with and initial condition

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A population

P(t)

satisfies

\dfrac{dP}{dt}=kP\,(500-P)

with

k>0

and initial condition

P(0)=600

. Which statement best describes the behavior of

P(t)

for

t>0

? (a)

P(t)

decreases toward 0 because the rate of change is negative for all positive

P

. (b)

P(t)

increases without bound because

kP(500-P)

is always positive when

P>0

. (c)

P(t)

decreases toward 500 because the rate of change is negative when

P>500

. (d)

P(t)

oscillates about 500 because the sign of

\dfrac{dP}{dt}

alternates as the population crosses

500