Query 4

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course MTH 279

2/4 5 pmI will be completely honest with you; I had no idea how to even tackle this section. Modeling has always been a struggle for me. I wish I could tell you what I didn't understand in each question, but I’m at a complete loss in almost all of these scenarios of how to even start out. In general, I don’t know how to apply the given information to the equation model. I tried reading the book, but it’s hard for me to get anything useful out of it. I also watched videos from the DVD's. If I had more time, I'd probably look through the class notes a bit. If you have any tips as to how I could organize the information in problems like this or maybe a method for setting up the problems, I would greatly appreciate it. I may be better able to understand it.

query 04

2.5.

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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Your solution:

93.75 gal/hr

Q’ = r * .03 - r * (50/1000)

-15/8 = -.02r

r = 93.75

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The concentration in the tank does not remain at 5%; that is, the quantity of salt in the tank does not remain at 50 gallons.

Q ' is the rate of change of the amount of salt in the tank. The amount of salt in the tank must therefore be Q.

If you replace that static 50 with the dynamic Q, you'll get an equation you can easily solve.

50 is the initial number of gallons of salt in the tank, which gives you the initial condition Q(0) = 50.

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: 2. Solve the preceding question if the tank contains 500 gallons of 5% solution, and the goal is to achieve 1000 gallons of 3.5% solution at the end of 8 hours. Assume that no solution is removed from the tank until it is full, and that once the tank is full, the resulting overflow is well-mixed.

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Your solution:

No difference.

@&

There is a difference. The tank only contains 500 gallons of solution at the initial instant, so no salt leaves the tank until it is full.

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

I’m sure there is a difference, but I don’t know how having the 500 gallons initially and 1000 gallons at the end would affect the solution to the problem, or how you would factor that into the equation.

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Figure out how much salt will be in the tank when it reaches 1000 gallons, and how many of the 8 hours you then have left. From that point on the equation is very similar to the preceding, but with slightly different conditions.

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Self-critique rating: 3

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Question: 3. Under the conditions of the preceding question, at what rate must 3% solution be pumped into the tank, and at what rate must the mixed solution be pumped from the tank, in order to achieve 1000 gallons of 3.5% solution at the end of 8 hours, with no overflow?

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Your solution:

41.6 gal/hr

Q’ = r_i * c_i and no outflow makes r_o * c_o be zero

10/8 = r * .03

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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Question: 4. Under the conditions of the first problem in this section, suppose that the overflow from the first tank flows into a large second tank, where it is mixed with 3% saline solution. At what constant rate must the 3% solution flow into that tank to achieve a 3.5% solution at the end of 8 hours?

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Your solution:

No attempt

@&

The first equation gives you the Q(t) function for the first tank. Knowing Q(t) and the outflow rate you can find the rate at which salt enters the second tank, as a function of t.

The second tank is large, so it simply accumulates the inflow from the first tank, and from the 3% solution flowing in.

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

In this problem, I’m assuming the c_i is changing constantly as it is added to the second tub, but I don’t know how to factor that into the equation.

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Self-critique rating: 3

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Question: 5. In the situation of Problem #1, suppose that solution from the first tank is pumped at a constant rate into the second, with overflow being removed, and that the process continues indefinitely. Will the concentration in the second tank approach a limiting value as time goes on? If so what is the limitng value? Justify your answer.

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Now suppose that the flow from the first tank changes hour by hour, alternately remaining at a set constant rate for one hour, and dropping to half this rate for the next hour before returning to the original rate to begin the two-hour cycle all over again. Will the concentration in the second tank approach a limiting value as time goes on? If so what is the limiting value? Justify your answer.

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Answer the same questions, assuming that the rate of flow into (and out of) the tank is 10 gallons / hour * ( 3 - cos(t) ), where t is clock time in hours.

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Your solution:

Yes, the limiting value is .03. If you solve the differential equation for Q and let t go to infinity, you get a maximum amount of salt to be 30 lbs. Since the total volume of the tank remains constant at 1000 gal, you can divide 30/1000 to get the maximum concentration.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

I didn't know how to take the fluctuating rate into consideration for the subsequent parts.

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Self-critique rating: 3

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Question: 6. When heated to a temperature of 190 Fahrenheit a tub of soup, placed in a room at constant temperature 80 Fahrenheit, is observed to cool at an initial rate of 0.5 Fahrenheit / minute.

If at the instant the tub is taken from the oven the room temperature begins to fall at a constant rate of 0.25 Fahrenheit / minute, what temperature function T(t) governs its temperature?

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Your solution:

No attempt.

@&

The rate at which temperature changes is proportional to the difference between the temperature of the soup and the temperature of the room.

That is,

rate of change = constant * difference in temperature

What is an expression for the temperature of the room as a function of time?

What is an expression for the difference in temperatures?

What is the value of the constant?

What therefore is the equation?

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

I did not know how to model the cooling equation when the temperature of the room isn’t constant.

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Self-critique rating: 3

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Question: "

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I've inserted suggestions as well as questions intended to break the solution process into smaller steps. See what sort of progress you can make, without an unreasonable expenditure of time, and submit a revision according to instructions below. I believe you'll be able to make reasonable progress. In the meantime, additional related or unrelated questions are welcome if they come up.

&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).

Be sure to include the entire document, including my notes.

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