Mixture Problem (pls help)

[johnglicer]

Homework Statement

A tank contains 80 gallons (gal) of pure water. A brine solution with 2 lb/gal of salt
enters at 2 gal/min, and the solution exits at 2 lb/gal. Find the amount of salt in the
tank at any time...

R1=C1Q1(c=concentration q=liquid substance or water ) this is the entering state
C1=2 lb/gal
Q1= 2gal/min

R2=C2Q2(same as above but) this is the exit state
C2= 2lb/gal
Q2= unknown

v= 80gallons

Homework Equations

dx/dt = R1 - R2
dx/dt = C1*Q1 - C2*Q2

but C2= x / vf
vf= v+ (Q1-Q2)t

The Attempt at a Solution

I tried to "alterate the formula or relevant equation given

of C2= x/vf to Q2= x/vf since i can't figure it out
then vf= 80+(2-0)t ;since q2 is not given(is this proper?)
vf=80+t
then
C2= x/(80+t)
dx/dt = R1 - R2
dx/dt = C1*Q1 - C2*Q2
dx/dt= 2(2) - 2x/(80+t)
dx/dt = 4 - 2x/(80+t) ;this is were I stuck up;; thought that it was separable

but I knew from the start that altering the equation is completely insane,, please help me out guys,,, if only C2 is not given then Q2 shows,, this would be easy...

please help me guys!


An image is provided just in case to sum it up..

 

[mighty2000]

Hello,

I understand that you are trying to find the amount of salt in the tank at any time, given the initial conditions and the rate at which the brine solution enters and exits the tank. Your approach seems to be on the right track, but there are a few things that need to be clarified and corrected.

Firstly, your expression for the volume of the tank (vf) is incorrect. It should be vf = v + (Q1 - Q2)t, where v is the initial volume of the tank (80 gallons), Q1 is the rate at which the brine solution enters the tank (2 gal/min), and Q2 is the rate at which the solution exits the tank (unknown). This expression represents the volume of the tank at any given time t, taking into account the initial volume and the changes in volume due to the inflow and outflow of the brine solution.

Secondly, you seem to have confused the concentration and the amount of salt in the tank. The concentration of the brine solution is given by the ratio of the amount of salt to the volume of the solution, and it is constant throughout the tank. On the other hand, the amount of salt in the tank is a function of time, and it changes as the brine solution enters and exits the tank. Therefore, your expression for C2 should be C2 = x/vf, where x is the amount of salt in the tank at any given time t, and vf is the volume of the tank at that time.

Now, using these corrections, we can write the differential equation for the amount of salt in the tank as follows:

dx/dt = R1 - R2
dx/dt = C1*Q1 - C2*Q2
dx/dt = (2 lb/gal)*(2 gal/min) - (x/(v + (Q1 - Q2)t))*(Q2)
dx/dt = 4 lb/min - (x/(80 + (2 - Q2)t))*(Q2)

This is a separable differential equation, and you can solve it using standard techniques. I hope this helps! Let me know if you have any further questions.