Question

A tank contains 1000 L of brine with 15kg of dissolved salt. Pure water enters the tank at a rate of 10L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank (a) after t minutes and (b) after 20 minutes?

Answer 1

I hate word problems

Answer 2

call the concentration of salt in the water \(Q(t)\) then we have\[\frac{dQ}{dt}=\text{rate at which Q(t) enters the tank}-\text{rate at which Q(t) exits the tank}\]\[=\text{rate of flow in}\times\text{concentration}-\text{rate of flow out}\times\text{concentration}\]

Answer 3

the concentration going in is zero, so we are left with\[\frac{dQ}{dt}=-\text{rate flow out}\times\text{concentration out}=-10\frac{Q(t)}{1000}\]and we have the initial condition that\[Q(0)=15/1000\]

Answer 4

\[\frac{dQ}{dt}=-10\frac{Q(t)}{1000}\]
\[dQ=\frac{-10}{1000} Q(t) dt\]
...and now take the integral right?