Assignment 6

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course Mth 279

7/3I don't understand this section at all. It seems that it all went right over my head even after watching the lectures, looking in the book, and trying to get outside help.

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:

1. input rate: 0.03r

output rate of salt qr/v = qr/1000

Rate of change of q is dq/dt = r_in - r_out = 0.03r - qr/1000

2. dq/dt = (r/1000)(30 - q)

1/(30-q) dq = r/1000 dt

integrate

ln(30-q) = rt/1000 + C

30 - q = Ce^(rt/1000)

q(0) = 50

30 - 50 = Ce^0

C = -20

q(t) = -20e^(rt/1000) + 30

3. when t = 8

output rate: qr/1000 ; q=r*1000 = .035*1000 = 35

q(t) = -20e^(rt/1000) + 30

35 = -20e^(8r/1000) + 30

5/20 = -e^(8r/1000)

ln(1/4) = - 8r/1000

r = ln(1/4) * -1000/8 = 173 gal/hour

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Given Solution:

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Self-critique (if necessary):

I am not confident on this section. I don’t understand exactly what I am doing and I don’t understand the problem. I had to look up how to solve this problem.

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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:

There is no question being asked here??? Not sure what I am to solve for?

Heres what I know

Start with V = 500 gal with 5% solution

want 1000 gal at t=8hr of 3.5%

so we would start with r = t/v

t = 500/r

I am not sure if this is the right direction for this problem. If so I am not sure where go next.

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The question is the same as on the preceding, but the conditions have changed.

How therefore would the preceding solution be modified for the new conditions?

You will want to break the problem into two parts, one representing what happens before the tank becomes full (this part does not require a differential equation) and one representing what happens after the tank is full.

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Given Solution:

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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:

Seeing that I was unable to answer the last question I don’t think I will be able to answer this question. Could you please provide the solution so I can try and walk myself through it to try to understand what to do and what it is asking???

q(8) = 1000

r_1 = flow input

r_2 = flow output

500(r_1 - r_2) = q(t)

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Nothing in your equation represents the rate at which the quantity of salt is changing. There is a rate of inflow of salt, and a rate of outflow.

You need to focus on the amount of salt in the tank, and on the rate of change of that amount. The rate of change will be related to the (changin) concentration of salt in the tank, the concentration of inflowing salt, and the rates of inflow and outflow.

Note that the rates of inflow and outflow have to bring the contents of the tank to 1000 gallons just at the end of the 8 hours.

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Given Solution:

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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:

In the same boat as the last question. Again this section with the word problems really seems to be tripping me up.

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You'll want to get the first few problems before tackling this one.

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Given Solution:

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Self-critique (if necessary):

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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:

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Given Solution:

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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:

T’(t) = k[80-T(t)]

integrate

T(t) = 80 + Ce^-kt

190 = 80 + Ce^(-0k)

190 = 80 + C

C = 110

T(t) = 80 + 110e^(-kt)

T’(t) = k(80 - T(t))

.5 = k(80-190)

k = .5/-110

T(t) = 80 + 110e^(.5t/-110)

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Good solution.

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Given Solution:

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Self-critique (if necessary):

Unsure where the .25F/min comes into play. I am not sure that I even solved this problem correctly.

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In this case the 80 F temperature will become the function T_room (t) = 80 F - .25 F/min * t. Replace 80 F in your equation with 80 - .25 t and solve.

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You did well on the first and last problems (but check my note regarding the second part of the last question).

You followed the procedure on the first problem; like most of us when you first work a problem you probably imitated a given solution without completely understanding what you did.

The remaining problems require you to understand what everything means, so you can adapt your solution.

I've inserted some notes to help guide you.

Additional questions are welcome. If you submit them, be sure to include everything, including my notes and your original solution, and denote your insertions by &&&&.

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