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

The background information for this problem was taken from:

https://www.tsijournals.com/articles/elzaki-transform-for-two-tank-mixing-problems.pdf

You do not need to know anything that appears in that pdf, the two functions will be defined for you, this is just for reference.

We  have  two 100 gallon tanks.  Tank A  is  initially  full  of  salt  water  with  a  salt  concentration  0.5  liter/gallon. Tank B  is  full  of  pure  water.  Pure  water  enters tank A  at  2  gal/min  water  and exits  the  well  mixed  tank  A  at  two  gal/min  and then flows into tank B.  The well mixed water in Tank B exits  at a rate of 2 gal/min and flows out of the system.   It can be shown that in this situation, the amount of salt in each tank can be modeled by the following two functions.

The amount of salt in Tank A (measured in liters), as a function of t (in minutes), is given by:
A(t) = 50(e^(-t/50))

The amount of salt in Tank B (measured in liters), as a function of t (in minutes), is given by:
B(t) = 50t (e^(-t/50))

A[t_]:=50*Exp[-t/50];
B[t_]:=t*Exp[-t/50];

Part 1

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

Part 2

Plot A(t) and B(t) on the same graph, for the time range t=0 minutes to t=200 minutes.

Part 3

Explain how the graphs given in part 2 agree with your intuition 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 4

When do both tanks have the same salinity?

Part 5

At what time was there the most salt in Tank B?  And how much salt was there?

Part 6

Compare your results from Part 4 and Part 5, and then explain.

Problem 2

Normally, if we draw two random lines in a plane, they will intersect at a single point.  What happens if we take three lines and plot them.  What do you think the chances are of all three points intersecting at single point?

Part 1

Consider the following two lines:
y=6x+5
y=-3x+4
Find where these two lines intersect by using the Solve command.

Part 2

Plot both of the lines, and verify that your solution is indeed where the two lines intersect.

Part 3

Verify that the following three lines do not have a single point of intersection.
y=6x+5
y=-3x+4
y = -x+1

Part 4

Find a value for b such that the following three lines do intersect at a single point.
y=6x+5
y=-3x+4
y = -x+b

Part 5

Plot all three lines on the same graph to verify that they all intersect at the same point.

Problem 3

The following table gives the total number of people fully vaccinated in Oklahoma over a window of time in September of 2021:

({{Day (in September),  People Fully Vaccinated }, {7, 1768580}, {8, 1776861}, {9, 1781639}, {10, 1788373}, {11, 1794448}, {12, 1801334}, {13, 1805668}, {14, 1808630}, {15, 1814425}, {16, 1820228}, {17, 1826304}, {18, 1832097}, {19, 1838725}, {20, 1842711}, {21, 1845303}, {22, 1849881}, {23, 1855558}})

Data was gathered from: https://github.com/owid/covid-19-data/tree/master/public/data/vaccinations

VaccineData =  {{7, 1768580},{8,1776861},{9,1781639},{10,1788373},{11,1794448},{12,1801334},{13,1805668},{14,1808630},{15,1814425},{16,1820228},{17,1826304},{18,1832097},{19,1838725},{20,1842711},{21,1845303},{22,1849881},{23,1855558}};

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 day from the data set was there the greatest increase in vaccinations?

Part 4

What is the average rate of change in the number of people vaccinated over the entire period of time in the data set.

Part 5

There were 1,859,672 fully vaccinated Oklahomans on September 24th.  Is this number greater or less than expected based on the rate of change from September 22nd to September 23rd?

Problem 4

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 VaccineData from Problem 3.  Look up “Interpolation” in the Help menu.

Part 2

Evaluate the interpolating function from Part 1 at 24 (i.e. if you called the function from part 1 F, evaluate F[24]).  This will be the approximation to the total number of Oklahomans full vaccinated by September 24th.

Part 3

Use the rate of change from September 22nd to the 23rd from Problem 4 to estimate the value of fully vaccinated people on the 24th.

Part 4

The answers to Parts 2 and 3 are not equal, so what can you conclude about the interpolating function with regards to average rates of change?