Question

a tank contains 20kg of salt dissolved in 5000L of water.Brine contains 0.3kg of salt per L and enters the tank at 25L/min. How much salt is in the tank after 30 mins if it drains at 25L/min?

Answer 1

@SolomonZelman help

Answer 2

@dumbcow hlp

Answer 3

set up differential equation

rate the salt changes = inflow of brine - outflow

\[\frac{ds}{dt} = 25 (.3) - 25 (\frac{s}{5000})\]
\[\frac{ds}{dt} = 7.5 - \frac{s}{200}\]
\[\frac{ds}{dt} = \frac{1500 -s}{200} \]
separate variables and integrate
\[\int\limits \frac{ds}{1500-s} = \int\limits \frac{dt}{200}\]
\[- \ln (1500-s) = \frac{t}{200} + C\]
\[1500 - s = C e^{-t/200}\]
\[s(t) = 1500 - C e^{-t/200}\]
plug in initial value of s(0) = 20
C = 1480
\[s(t) = 1500 -1480e^{-t/200}\]
Finally plug in t=30 to get answer