A tank contains a brine solution whose concentration is given by

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

C(t)=\dfrac{m(t)}{V(t)}

, where

m(t)

is the mass of salt (in grams) and

V(t)

is the volume of solution (in liters). At a certain instant, the mass of salt remains constant while the solution continues to drain so that

\dfrac{dV}{dt}<0

. Which of the following best describes the sign of

\dfrac{dC}{dt}

at that instant? (a)

\dfrac{dC}{dt}=0

because

\dfrac{dC}{dt}=\dfrac{dm/dt}{dV/dt}

and

dm/dt=0

. (b)

\dfrac{dC}{dt}>0

because

\displaystyle\dfrac{dC}{dt}=-\dfrac{m}{V^{2}}\dfrac{dV}{dt}

and

\dfrac{dV}{dt}<0

. (c)

\dfrac{dC}{dt}<0

because applying the quotient rule as

\dfrac{dC}{dt}=\dfrac{m}{V^{2}}\dfrac{dV}{dt}

gives a negative value when

\dfrac{dV}{dt}<0

. (d) The sign of

\dfrac{dC}{dt}

cannot be determined without knowing how fast the volume is changing.