A well-stirred mixing tank contains a brine solution whose mass of dissolved salt is…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A well-stirred mixing tank contains a brine solution whose mass of dissolved salt is

m(t)

kilograms and whose total volume of liquid is

V(t)

liters, both varying with time

t

(measured in minutes). The salt concentration in the tank is defined by

c(t)=\frac{m(t)}{V(t)}.

At a certain instant the following measured data are recorded: • mass of salt:

m=30\,\text{kg}

• volume of solution:

V=120\,\text{L}

• rate of change of mass:

\dfrac{dm}{dt}=0.50\,\text{kg min}^{-1}

• rate of change of volume:

\dfrac{dV}{dt}=1.2\,\text{L min}^{-1}

Answer the questions that follow. (a) Derive a formula for

\dfrac{dc}{dt}

in terms of

m

V

\dfrac{dm}{dt}

, and

\dfrac{dV}{dt}

. (b) Using the recorded data, calculate

\dfrac{dc}{dt}

at the given instant. Include the correct units in your final answer. (c) Interpret the sign of

\dfrac{dc}{dt}

in the context of the salt concentration in the tank.