Water Tank Differential Equation Problem

[harrietstowe]

Homework Statement

A tank initially contains 400 gal of brine in which 100 lb of salt are dissolved. Pure water is run into the tank at the rate of 20 gal/min, and the mixture (which is kept uniform by stirring) is drained off at the same rate. How many pounds of salt remain in the tank after 30 minutes?

Homework Equations

The Attempt at a Solution

These were my variable definitions:
Let y = gallons of brine
y_o=400 gal
d_y/d_t=-20 gal/min
Let w= gallons of water
w_o= 0 gallons
d_w/d_t= +20 gal/min
y=y_o+(d_y/d_t)t
y=400gal+(-20gal/min)(30min)
w=w_o+(d_w/d_t)(t)
I was concerned because I got a negative answer
The physics side of me says that the water that is being poured in should cause the brine to leave the tank at an even faster rate but that would just give me an even more negative answer.
Thanks

 

[micromass]

Well, your mistake is in saying that dy/dt=-20. This is true in the beginning. But suppose that after a certain time, the mixture in the tank is 50% brine and 50% water, then (at that time) dy/dt=-10. Thus the amount of brine that is poured away every minute decreases. Your ODE should model that...

 

[harrietstowe]

ok can you take that idea a little further please? I mean get what your saying but I am struggling to express that idea in the form of an equation

 

[micromass]

Well, at time t, there is y(t) gallons of brine in the water. Thus the concentration of brine in the water is y(t)/400. Since the water is flowing of at 20 gal/min, the amount of brine that pours out at time t is 20y(t)/400. Thus your differential equation is dy/dt= 20y/400.