Question

A tank initially holds 10 lit. of fresh water. At t = 0, a brine solution containing $\frac{1}{2}$ kg of salt per lit. is poured into tank at a rate 1 lit/min while the well-stirred mixture leaves the tank at the same rate. Find the amount of salt in a tank at a particular time

• A. $-5e^{-t}+5$
• B. $5e^{-t}+5$
• C. $5e^{-t}-5$
• D. $-5e^{}-5$

Answer 1

Answer: (A)

$-5e^{-t}+5$

Let the concentration of salt in the tank at time $t$ be $c$
then by mass balance of salt for time $dt$

$\frac{dC}{dt}=0.5-c$ [rate at which mixture leaves the tank]

Integrating
$\ln (0.5-c)=-t+k$ using formula $\int \frac{du}{u}=log|u|+C$
given at $t=0$, $c=0$,

hence

$k=-\ln 2$
$c=0.5(1-e^{-t})$
Amount of salt in the tank at time t
$=c\times 10$
$=5(1-e^{-t})$