#$&*
Mth 279
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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Question about Query 04 (A#6)
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1. A 3% saline solution flows at a constant rate into a 1000-gallon tank initially full of a 5% saline solution. The solutions remain well-mixed and the flow of mixed solution out of the tank remains equal to the flow into the tank. What constant rate of flow is necessary to dilute the solution in the tank to 3.5% in 8 hours?
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I think I used the given information in the problem correctly in developing an equation about rate of change, however after this step I am struggling to solve the differential equation in terms of the inflow rate rather than in terms of Q at time t which is how it is solved in the book example. Here is what I have so far:
Rate of change of salt in tank=r_1(t)*c_1(t)-r_0(t)[Q(t)/V(t)]
dQ/dt= r_1(t)*c_1(t)-r_0(t)[Q/V(t)], Q(0)=Q_0=0.05*(1000gals)=50lbs of salt
dQ/dt=(r_1(t))(0.03)-(r_1(t))[50lbs/1000gals]
dQ/dt=(r_1(t))(0.03)-(r_1)(t))(0.05)
dQ/dt=r_1(t)(-0.02)
I'm stuck on solving for r_1. Unfortunately all of the succeeding problems are dependent on this as well. Thanks for the help.
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@& The rate of inflow is constant, so you can call this rate r.
The tank is initially full so outflow will also be at rate r.
Your task is to solve for the rate.
.03 r is the rate at which salt enters the tank.
Outflow concentration is Q / V. V is constant at 1000 gallons. Q is changing as the less concentrated inflowing water is mixed with the more concentrated water in the vat.
You treat Q as if it is a constant 50 lbs, rather than a function of t.
At t = 0 you have Q = 50 lbs. After that Q is less than 50 lbs. (Q will be approaching 30 lbs, which you should be sure you understand intuitively).
See if this gets you on track, and if you're still bogged down after a reasonable effort, please ask again. If you do, be sure to include my response and your original work so I can see what you've did originally, what I told you, and what you've done since.*@