Two connected tanks contain positive amounts of a dye

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

Problem

Two connected tanks contain positive amounts of a dye, measured in grams, at time

t

. Let

A(t)

and

B(t)

denote the masses in tank A and tank B, respectively. The rates satisfy

\frac{dA}{dt}=k_{1}B, \qquad \frac{dB}{dt}=k_{2}A,

where

k_{1}

and

k_{2}

are non-zero constants. At

t=0

the masses satisfy

A(0)>0

and

B(0)>0

, and observations show that

A

is increasing while

B

is decreasing at that instant. Which set of signs for

(k_{1},k_{2})

is most consistent with the observations? (a)

k_{1}<0,\;k_{2}<0

(b)

k_{1}>0,\;k_{2}>0

(c)

k_{1}>0,\;k_{2}<0

(d)

k_{1}<0,\;k_{2}>0