A brine is a solution of salt in water. If the brine is in a tank equipped with fill and drain pipes, then the total amount of dissolved salt in the tank varies as the salt concentration in the inflow stream changes and the inflow and outflow rates are adjusted. The amount of salt in the tank can be modeled by using the following law.
Balance Law: Net rate of change = rate in – rate out
It is assumed throughout that the inflow stream is instantaneously mixed with the brine in the tank so that at any given time the concentration of salt in the tank is uniform.
A tank with a capacity of 4000 liters holds 2000 liters of brine that contains 50kg of dissolved salt. Brine with a salt concentration of 0.2kg per liter is piped into the tank at a rate of 40 liters per minute. Well mixed brine is drawn off at k liters per minute. The model for the amount x(t) of salt in the tank at time t is given by:
dx/dt=(0.2)(40) - (x/(2000+(40-k)t))(t)
Rewrite the differential equation in linear form, find an integrating factor, and find a formula for the amount x(t) of dissolved salt as a function of time t and outflow rate k.
I tried this several times and couldn't get it, so if anyone could help me with it that'd be great. Thanks!
Please show all your work and say where you get stuck.