Math 1483 - Quiz 4

YOUR NAME HERE

Remember, ALT+7 gives a new text cell, be sure to explain your answers, if needed!

Problem 1

We  have  two 100 gallon tanks, call them tank A and tank B.  Tank A has 50 lbs of salt dissolved in it, and tank B has 300 lbs of salt in it.  The tanks are connected by two tubes, the first pumps water from tank A to tank B at a rate of 5 gallons/minute, while the second moves water from tank B to tank A at 2 gallons/minute.  Furthermore, water is pumped into tank A from an outside source at a rate of 3 gallons/minute, and the salinity of this water is 1 lb of salt/gallon.  Tank B has a spigot which is open, and water pours out of it at a rate of 5 gallons/minute.


The amount of salt in Tank A (measured in liters), as a function  of t (in minutes), is given by:
A(t) = -25 (e^(-(t/20)-(t/(10 (Sqrt[10])))))-20 (Sqrt[10]) (e^(-(t/20)-(t/(10 (Sqrt[10])))))-25 (e^(-(t/20)+(t/(10 (Sqrt[10])))))+20 (Sqrt[10]) (e^(-(t/20)+(t/(10 (Sqrt[10])))))+100 (e^(-(t/20)+(t/(10 (Sqrt[10])))-(1/100) (-5+(Sqrt[10])) t))

The amount of salt in Tank B (measured in liters), as a function of t (in minutes), is given by:
B(t) = 100 (e^(-(t/20)-(t/(10 (Sqrt[10])))))+25 (Sqrt[(5/2)]) (e^(-(t/20)-(t/(10 (Sqrt[10])))))+100 (e^(-(t/20)+(t/(10 (Sqrt[10])))))-25 (Sqrt[(5/2)]) (e^(-(t/20)+(t/(10 (Sqrt[10])))))+100 (e^(-(t/20)+(t/(10 (Sqrt[10])))-(1/100) (-5+(Sqrt[10])) t))

A[t_]:=-25 (E^(-(t/20)-(t/(10 (Sqrt[10])))))-20 (Sqrt[10]) (E^(-(t/20)-(t/(10 (Sqrt[10])))))-25 (E^(-(t/20)+(t/(10 (Sqrt[10])))))+20 (Sqrt[10]) (E^(-(t/20)+(t/(10 (Sqrt[10])))))+100 (E^(-(t/20)+(t/(10 (Sqrt[10])))-(1/100) (-5+(Sqrt[10])) t));
B[t_]:=100 (E^(-(t/20)-(t/(10 (Sqrt[10])))))+25 (Sqrt[(5/2)]) (E^(-(t/20)-(t/(10 (Sqrt[10])))))+100 (E^(-(t/20)+(t/(10 (Sqrt[10])))))-25 (Sqrt[(5/2)]) (E^(-(t/20)+(t/(10 (Sqrt[10])))))+100 (E^(-(t/20)+(t/(10 (Sqrt[10])))-(1/100) (-5+(Sqrt[10])) t));

Part 1

Evaluate A(0) and B(0) and compare those results to the situation described in the problem.  Do they agree?

Part 2

So no peeking ahead at part 3 here.  Clearly the functions which define A(t) and B(t) are pretty complicated looking.  However, if you look at the problem and consider how the tanks are set up, how much salt should each tank have in it if this system is allowed to run for a very long time.  Note: you will NOT be marked off for an incorrect answer here, you just need to think about the problem and give a justification for what you think will happen.

Part 3

Plot A(t) and B(t) on the same graph, for the time range t=0 minutes to t=100 minutes.

Part 4

Explain how the graphs given in part 3 agree (or disagree) with your intuition (part 2) about how much salt should be in each tank as time passes.  Be sure to include the initial state, the mixing of the tanks, and the amount of salt in the tanks after a long period of time has passed.

Part 5

Modify your graph from part 3 to include the limiting value each function tends to.

Part 6

Do the two tanks ever have the same salinity (lbs of salt/gallon)?

Part 7

Create a three-column table with the values of t, A(t), and B(t) for 0<= t <= 200 minutes by increments of 10 minutes.  Be sure to express entries in the the columns corresponding to A(t) and B(t) in decimal form.

Part 8

Is there a time at which tank A has the limiting value of 100 lbs of salt in it?

Part 9

Is there a time at which tank B has the limiting value of 100 lbs of salt in it?

Part 10

At what time does tank A have a maximum amount of salt?

Part 11

At what time does tank B have a maximum amount of salt?

Part 12

On what time interval is the amount of salt in tank B between 150 and 250 lbs?

Problem 2

The following table gives the price of gas (per gallon) for each week of the calendar year 2022.

({{Week of 2022,  Price/Gallon }, {1, $3.281}, {2, $3.295}, {3, $3.306}, {4, $3.323}, {5, $3.368}, {6, $3.444}, {7, $3.487}, {8, $3.530}, {9, $3.608}, {10, $4.102}, {11,  $4.315}, {12, $4.239}, {13, $4.231}, {14, $4.170}, {15, $4.091}, {16, $4.066}, {17, $4.107}, {18, $4.182}, {19, $4.328}, {20, $4.491}, {21, $4.593}})

Data was gathered from: https://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=pet&s=emm_epmr_pte_nus_dpg&f=w)

GasData =  {{1, 3.218},{2,3.295},{3,3.306},{4,3.323},{5,3.368},{6,3.444},{7,3.487},{8,3.530},{9,3.608},{10,4.102},{12,4.315},{13,4.239},{14,4.231},{15,4.170},{16,4.091},{17,4.066},{18,4.182},{19,4.328},{20,4.491},{21,4.593}};

Part 1

Plot the data.  Be sure to label your axes appropriately.

Part 2

Compute the rate of change for the data.

Part 3

On what week from the data set was there the greatest increase in gas price nationally?

Part 4

What is the average rate of change in the price per gallon of gas over the entire first 21 weeks of this year.

Part 5

When I was writing this exam, the date was May 25th, 2022 (week 21).  Thus, data for the work week starting May 30th, 2022 was not yet available(week 22).  Using the rate of change from week 20 to week 21, estimate the value of the price of gas for the next data point, May 30th, 2022, which is week 22.

Part 6

Go to the website linked at the beginning of this problem, and compare the actual value with that of the estimate from part 5.  Explain in detail whether or not you think the estimate was a good estimate.

Problem 3

Time for a “Let’s find something fun to do with Mathematica” problem.

Part 1

Have Mathematica create an interpolation function based on the data set GasData from Problem 2.  Look up “Interpolation” in the Help menu.

Part 2

Evaluate the interpolating function from Part 1 at 22 (i.e. if you called the function from part 1 F, evaluate F[22.0]).  This will be the approximation to the price of gas/gallon for week 22 of the 2022 year.

Part 3

Compare your result from part 2 to part 5 of the previous problem.  Do they agree? What can you conclude about the interpolating function with regards to average rates of change?