Answer:
The amount of salt in the tank at any moment is
The concentration of salt in the tank when it is at the point of overflowing is .
The theoretical limiting concentration of an infinite tank is lb per gallon.
Step-by-step explanation:
Let be the amount of salt in the tank at any time
Then, its time rate of change, , by (balance law).
Since three gallons of salt water runs in the tank per minute, containing lb of salt, the salt rate is
The amount of water in the tank at any time is.
Now, the outflow is gal of the solution in a minute. That is
of the total solution content in the tank, hence
of the salt salt content
, that is
.
Initially, the tank contains lb of salt,
Therefore we obtain the initial condition
Thus, the model is
Linear ODE.
so, an integrating factor is
and the general solution is
Now using the initial condition and find the value of .
is the amount of salt in the tank at any moment
Initially, the tank contains gal of water and the capacity of the tank is
gal. This means that there is enough place for
gal
of water in the tank at the beginning. As concluded previously, we have one new gal in the tank at every minute. hence the tank will be full in min.
Therefore, we need to calculate to find the amount of salt any time prior to the moment when the solution begins to overflow.
To find the concentration of salt at that moment, divide the amount of salt with the amount of water in the tank at that moment, which is L.
If the tank had an infinite capacity, then the concentration would be
Hence, the theoretical limiting concentration of an infinite tank is lb per gallon.