Linear Algebra System of Equations/Rates Application Help

[bran_1]

Homework Statement

Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains
x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of salt in 300 gallons of brine. The mixture in each tank is kept uniform by constant stirring. When t = 0, brine is pumped from tank A to tank B at 20 gallons/minute and from tank B to tank A at 20 gallons/minute. Find the amount of salt in each tank at time t if x(0) = 10 and y(0) = 40.

Homework Equations

I know how to solve the system, but I'm having trouble setting up the intial equations from the conditions given.

The Attempt at a Solution

(1/200)dx(t)/dt = -x(t)/20 + y(t)/20
(1/300)dy(t)/dt = x(t)/20 - y(t)/20

These two equations yield the wrong answer, so I know my setup is incorrect. I'm mostly confused as to how to setup 2 equations for x(t) and y(t) for salt when the stuff I've been given are in salt/gallon.

 

[andrewkirk]

The system is confused by having coefficients on both sides of each equation. It will be easier to make sense of with coefficients only on one side of each. dx(t)/dt and dy(t)/dt are the rates of change of the number of pounds of salt in tanks A and B respectively. Those are nice simple concepts so let's have them on the left-hand side of each equation by themselves.

Now on the right-hand side of each equation we want two terms, one for the rate at which salt enters and one for the rate at which it leaves. Since all flows are at the rate of 20 gal/min, all terms will be that 20 gal/min multiplied by a salt concentration measure, ie pounds per gallon. So to get the amount to put in each of the RHS terms you need an expression for the salt concentration at time t in each of the tanks, in terms of x(t) and y(t).