Mathematical Modeling
Dr. Molli Jones, PA3-MSP Year 3
Activity 1: Linear Models
http://www.nasa.gov/audience/foreducators/exploringmath/algebra1/Prob_ShuttleMassTime_detail.html
Scatter Plots and Best-Fit Lines
Determining Linear Equations
Modeling Data with Linear Functions
Activity 2: Open-Ended Modeling Problems
Modeling Cycle
Reading and Understanding Quantitative Information
Creating Mathematical Models
Presenting and Interpreting Results
Adapted from Don Small, Contemporary College Algebra: Data, Functions, Modeling, Sixth Edition
Activity 1: Linear Models
Excerpts from “Exploring Space Through Math: Space Shuttle Ascent – Mass v. Time”
Problem:On July 4, 2006 Space Shuttle Discovery launched from Kennedy Space Center on mission STS-121, to begin a rendezvous with the International Space Station, or ISS. Before each mission, the projected data is compiled to assist in the launch of the space shuttle to ensure safety and success during the ascent. To complete this data, flight design specialists take into consideration a multitude of factors such as space shuttle mass, propellant used, mass of payload being carried to space and mass of payload returning. They must also factor in atmospheric density, which is changing throughout the year. After running multiple tests, information is compiled in a table showing exactly what should happen each second of the ascent.
The Booster flight controller monitors system health for the Space Shuttle Main Engines (SSMEs), the Solid Rocket Boosters (SRBs), and the External Tank (ET). They monitor pressures, temperatures, propellant flow rates, and valve positions that show that the engines are running and controlling properly. Booster also monitors all the pipes and valves that move propellant from the ET to the SSMEs. Propellant flow rates are significant because they determine how mass changes over time which affects acceleration.
Table 1 shows the total mass of Discovery for mission STS-121 every 10 seconds from liftoff to SRB separation. Total mass includes the orbiter, SRBs, ET, propellant, and payload. It is during the first stage of the ascent, that the space shuttle is burning the greatest amount of propellant. You can see in the table that the space shuttle has a total mass of 2,051,113 kg at t = 0. After 2 minutes its total mass is only 880,377 kg, or 43 % of the original mass. The burning of this vast amount of propellant is needed to get the space shuttle through Earth’s atmosphere and into orbit.
Time (s) / Space Shuttle Total Mass (kg)
0 / 2,051,113
10 / 1,935,155
20 / 1,799,290
30 / 1,681,120
40 / 1,567,611
50 / 1,475,282
60 / 1,376,301
70 / 1,277,921
80 / 1,177,704
90 / 1,075,683
100 / 991,872
110 / 913,254
120 / 880,377
Table 1: STS-121 Discovery Ascent data (total mass)
1. Describe what you think the graph of the Ascent Data will look like.
2. Now, use the graph paper provided to create a scatter plot of the total mass v. time. Think carefully about the scale that you should use for the graph. Once you have your scatter plot, draw a line of best fit on your scatter plot. Use two points on this line to determine the equation of the line of best fit. How well does the line fit the data?
3. Based on the scatter plot, what is the correlation of the data (positive, negative, constant, or no correlation)? Explain this in terms of the variables in the problem.
4. Table 2 shows several familiar objects and the approximate mass in kilograms of each one.
a. To gain perspective regarding the magnitude of the propellant consumption of the space shuttle, find the approximate number of each type of object that it would take to equal the mass of the entire space shuttle system at launch. Round to the nearest whole number.
Object / Approximate mass (kg) / Approximate number of objects to equal the mass of the space shuttle at launchStatue of Liberty / 204,117
Boeing 747 airplane / 158,757
Fuel tank truck / 27,216
School bus / 11,340
Table 2: Mass of Various Objects
b. The slope of the best fit line that you found in question 2 is the amount of propellant in kilograms that is burned per second. If a Boeing 767 airplane burns about 24,500 kg of fuel on a 6 hour flight from New York to Los Angeles, about how much time would it take the space shuttle to burn an equivalent amount of propellant?
5. Consider the data set in Table 1. When time is zero seconds, the total mass shown in the table is the actual value of the mass at liftoff. The equation of the line of best fit that you found In question 2 models the data in Table 1. However, a model of any data set contains some error.
a. Use the equation for the line of best fit that you found in question 2 to find the estimated value of the mass in kilograms when t = 0 at liftoff.
b. Determine the percent error in the equation of the line of best fit. Round to the nearest tenth.
6. A second method for approximating the data would be to use the information that is given rather than trying to visually determine the line of best fit.
a. To determine an approximate slope, look at the slope between each pair of points and use this information to approximate the slope of the line.
Time (s) / Space Shuttle Total Mass (kg) / Change in Mass (kg/s)0 / 2,051,113 / ----
10 / 1,935,155 / (1,935,155 - 2,051,113)/(10 – 0) = -11,595.8 kg/s
20 / 1,799,290
30 / 1,681,120
40 / 1,567,611
50 / 1,475,282
60 / 1,376,301
70 / 1,277,921
80 / 1,177,704
90 / 1,075,683
100 / 991,872
110 / 913,254
120 / 880,377
Approximate slope =
b. Now, using the actual mass at liftoff, write an equation of a line that approximates the data. Graph this line. Which line gives the better approximation of the data? Why?
Activity 2: Open-Ended Modeling Problems
You will be assigned one of the projects. Near the end of the class period, each group will be given five minutes to present their work. All of the presentations must follow the modeling cycle. In particular, the presentation should include:
· A brief restatement of the problem or situation
· A list of the assumptions that were made in creating the model
· A description of the mathematical model that was created
· A description of the analysis that was performed
· The solution that was determined
· An interpretation of the results
· A description of how the results will be used to improve the model or to answer further questions
These projects are purposely left open-ended because that is how problems are presented in real-life. Do your best!
(A good source for problem ideas is the NASA PUMAS site – https://pumas.gsfc.nasa.gov/.)
Project 1: Harvesting Trout
White Elk State Forest
1600 Elk Road
Big Sky, MT 59600
August 2, 2012
PA3-MSP Mathematics Participants
Immaculata University
1145 King Road
Immaculata, PA 19345
Dear Mathematics Teachers:
This past spring, we were low on funds, so we started selling fishing licenses and allowing trout fishing in our main lake. Now we have a problem – the trout are dying off! The park needs the revenue from selling the licenses, so we need to figure out how to stabilize the situation. In particular, we need to decide how many licenses to sell and how many fish need to be stocked each spring in order to optimize our profits and keep the fish population at a reasonable level. In these hard economic times, our budget has been cut severely. We have some preliminary data, but we have been unable to hire anyone to analyze the data. Our park and our trout are counting on you!
Before we allowed any fishing in the lake, the trout population was decreasing by about 8% per year due to natural predators. This past September there were around 8000 trout in the lake. We sold 1000 licenses at $25 apiece. Based on what other parks charge for licenses, we will lose about 250 customers for every $5 we raise the price of a license. Now that the fishing season is over the fish population is around 2000. A trusted environmental organization has informed us that to keep from damaging the ecosystem, the trout population must stay between 5,000 and 10,000 fish. It costs about a dollar per fish to stock the lake.
As a park ranger, I do not have a strong mathematical background, so please try to avoid technical jargon. Thank you for your help.
Sincerely,
Ranger Tad Herring
Project 2: Saving Water
(Adapted from Don Small, Contemporary College Algebra: Data, Functions, Modeling, Sixth Edition)
1. Calculate how much water is used in your home per day. Provide a description and amount for each type of water usage.
2. Come up with a way to visualize this daily amount of water.
3. How much water is lost in one year through a faucet that drips every 30 seconds?
4. Compare the amount of water used for a shower as opposed to a bath.
5. Suggest other ways to save water and calculate how much water they would save.
Project 3: Forensic Mathematics
East Whiteland Township
Police Department
209 Conestoga Road
Frazer, PA 19355
August 5, 2012
PA3-MSP Mathematics Participants
Immaculata University
1145 King Road
Immaculata, PA 19345
Dear Mathematics Teachers:
As a detective for the East Whiteland Police Department, I am quite familiar with Immaculata University. We hope that you can help us on a recent case (that thankfully does not involve any Immaculata students).
On Friday, August 3, at 3:28 PM, a masked man walked into Penn Liberty Bank, located at 199 Lancaster Avenue, in Malvern, PA. The man held up a teller and made off with approximately $5000 in cash. We have arrested a suspect, but he is claiming an alibi. We would like you to determine if the alibi will hold up in court.
Security footage shows that the suspect left his place of employment, the McDonald’s located at the intersection of Route 100 and Swedesford Road, at 3:03 after a spat with the manager that ended with the immediate firing of the suspect. The suspect then claims that he stopped at Starbucks, located at 300 E. Lincoln Highway in Exton, PA, where he ordered a Doppio Espresso. The suspect claims that he could not have arrived at the bank by 3:28.
An additional piece of evidence that we collected was that witnesses have reported that shortly before the bank robber’s car pulled into the parking lot, the bank robber slammed on his brakes to avoid running a traffic light. We measured the skid marks he left on the road (which was made of asphalt), and the marks were approximately 135 feet long. This may help you determine how fast the driver was going along Lancaster Avenue (Route 30).
Please use your mathematical skills to determine if the suspect could have arrived at the bank if he left the McDonald’s at 3:03 and stopped to order coffee. Please be as thorough as possible since we would not want to wrongfully convict the suspect.
Thank you for your help.
Sincerely,
Detective S. Holmes
Project 4: Burning Coffee
(Adapted from Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go)
J.I.T. Box
132 Fast Food Lane
Partially Hydrogenated, WI 33021
August 1, 2012
PA3-MSP Mathematics Participants
Immaculata University
1145 King Road
Immaculata, PA 19345
Dear Mathematics Teachers:
After the verdict against one of our competitors a few years ago, my company, which owns a series of quick-dining establishments, has been hit with a series of copy-cat lawsuits. We need some technical expertise on one of these matters, and your enterprising and resourceful professor referred me to you.
The plantiff, whom I will refer to as R. Clumsy for legal reasons, was the passenger in an automobile that stopped at one of our drive-through windows early one morning for coffee. Mr. Clumsy placed the coffee in a cup holder for several minutes, held the coffee cup in his hands for a short time, and then he spilled the coffee. He maintains that the coffee was much too hot, and he is suing for $200,000 for emotional distress and dry cleaning bills.
I would like to know how credible his story is. While we do serve our coffee at 160○F, I do not believe that the coffee was above the industry standard of 140○F when Mr. Clumsy spilled his coffee. He claims that it was exactly 7:58 AM when the coffee was poured, and when he spilled the coffee, some of it fell on his watch, a cheap Timex ripoff, which stopped at precisely 8:08 AM. He says that he left the coffee in the cup holder for 5 minutes, and held it in his hands for another 5 minutes before spilling it. I would like to be able to raise a reasonable doubt about the credibility of the story, either by showing that the temperature was not above the industry standard, or by showing that if the times are slightly different than he claims, then the coffee was not too hot.