MATH 163 Fall 2022

Applications Project

Pre-Calculus I is a general education course designed to assist students in the development of critical life skills.  One of the goals of this assignment is to assess student competence for each of these objectives:

  1. Written and Oral Communication— articulate a solution to mathematical problems (CCO20),
  2. Critical Analysis and Reasoning— produce and compare graphs of functions, using transformations, symmetry, end behavior, and asymptotes (CCO16),
  • Technological Competence— apply appropriate technology to the solution of mathematical problems (CCO21),
  1. Scientific and Quantitative or Logical Reasoning— identify domain and range of functions (CCO4), and
  • Personal and Professional Ethics— construct a solution to real world problems using problem solving methods individually and in groups (CCO18).

 

ASSIGNMENT: 
Over the last decade, at least, there has been much talk about climate change.  There have been discussions concerning the role humans have played in the altering of Earth’s atmosphere and the real and potential impact on life as we have come to know it.  Finally, there have been attempts already made to stop or reverse any negative results that may have been or that may be caused as humans continue to produce and consume goods and services in our ever-expanding societies.

In this assignment you will have the opportunity to investigate four scenarios related to the issues climate change.

Purpose:  The purpose of this assignment is to provide you, the student, with an opportunity to demonstrate your level of mastery of the mathematical and logical concepts that are presented in this pre-calculus course.

Audience:  The audience for this assignment is your pre-calculus professor, or any individual with a sound knowledge of the topics covered in the questions posed along with each scenario.

Directions:  Please respond to TWO of the THREE scenarios with complete ideas and sentences.  Be clear and succinct in your submissions.  Also, make sure to provide all required technology displays or output.

 

ASSIGNMENT SPECIFICATIONS: 

PICK ONLY 2 out of the 3 problems below to complete.

  • Cut and paste all technology displays and outputs, as needed, directly into the final document.
  • Your project should be typed; however, mathematical calculations may submitted as photo within the document.
  • Please delete these directions on pages 1, 2, &3. Keep the project scenarios.  You will answer each part of the scenario directly below the question asked.
  • Both CCBC and Purdue University offer very good online references at: ·

http://www.ccbcmd.edu/Resources-for-Students/Tutoring-and-Academic- Coaching/Writing-Center- and-Online-Writing-Lab/Documenting-and-Citing-Sources.aspx

http://owl.english.purdue.edu/owl/resource/747/01/

 

SUBMISSION GUIDELINES:

  • Send electronically with your name and class to dtruszkowski@ccbcmd.edu

Instructions on how to use Desmos, Excel, and Calculators to sketch a graph of an equation, to make a table and plot points, and to find an equation that models your data (i.e., the linear regression line or the line of best fit)

  1. Desmos
  2. To create a new graph, go to comand just type your expression in the expression list bar. As you are typing your expression, the calculator will immediately draw your graph on the graph paper. You can graph a single line by entering an expression like y = 2x + 3.
  3. Finding an equation that best fits your data in Desmos
  4. Go to comand choose Start Graphing.
  5. Click the plus sign in the upper left and choose Add Item > table
  6. Type your data in the table.
  7. Click on the wrench in the upper right to change the graph settings.
  8. Modify your x, and y values to reflect your data.
  9. Adjust the values of the sliders until the graph of the equation most closely fits your data points.
  10. On the DESMOS Calculator, type and select all to access the sliders to adjust the slope m and y-intercept b. Please use the following links for more directions on how to use DESMOS to sketch a graph of an equation, to make a table and plot points, and to find an equation that fits your data. Below are online videos that will help with explanation. Watch them in order. Please copy and paste the following links to the google site to avoid any possible error.
  11. Helpful videos:

https://learn.desmos.com/tables

https://learn.desmos.com/graphing

https://www.youtube.com/watch?v=4NTf551hHQE

 

  1. Excel
  2. You can also use Excel to get a regression line (line that models your data). These are the steps:
  3. Open Excel and fill your data in two columns: 1st column representing the x coordinates and second column representing the y coordinates of your points.
  4. Select both columns by highlighting them.
  5. Go to insert and at the chart part of the menu, you will see the scatterplot. Click on it and select the first scatterplot option. A scatterplot of your data will appear on the screen
  6. You can either click on the “Add chart element” at the top left of the menu or just click on the “plus” sign at the top corner of the scatterplot. Check the trendline box. It will show the trendline (regression line). To add an equation for the trendline, click on the arrow next to the trendline option and it will show you “more options”. You scroll down until you see “display the equation” and you check that box. The equation of the trendline will appear on the screen. It is usually in a “small” font and a little hard to read. You can move it around the screen and change the font and make it bigger.
  7. The following link shows these steps: https://www.youtube.com/watch?v=L_a8Z0BVjyM

 

  • Calculators: How to find the linear regression line:

 

  1. Casio fx-115ES Plus
  2. Press Mode, then 3 for STAT, then 2 for A+BX.
  3. Input all of your independent data under the X column, and all of the dependent data under the Y column.
  4. Press AC to exit edit mode safely.
  5. Press SHIFT & 1 to enter STAT/DIST mode.
  6. Press 5 for Reg.

 

  1. Casio fx-991EX
  2. Press MENU, then 6 for Statistics, then 2 for y=a+bx.
  3. Input all of your independent data under the x column, and all of the dependent data under the y column.
  4. Press AC to exit edit mode safely.
  5. Press OPTN.
  6. Press 3 for Regression Calc. Everything is now on your screen, although you may need to scroll down to find relevant information.
  7. When you are finished, your calculator will still be in y=a+bx Statistics mode. Press MENU, then 1 to return to Calculator mode.

 

  1. TI-84 Plus or similar

Press STAT, then ENTER to input your data.

  1. Input all of your independent data under the L1 column, and all of your dependent data under the L2 column
  2. Press 2ND & MODE to quit out of edit mode.
  3. Re-Press STAT, but this time scroll over to CALC.
  4. Scroll down to LinReg(ax+b).
  5. Press ENTER to select LinReg(ax+b).
  6. Scroll down to highlight CALCULATE.
  7. Press ENTER to see all of the data. This will display the correlation, slope of line of best fit (called a), and the y-intercept of the line of best fit (called b).

 

Math 163 Applied Project

  • Carbon Dioxide Emissions

Carbon emissions contribute to climate change, which has serious consequences for humans and their environment. According to the U.S. Environmental Protection Agency, carbon emissions, in the form of carbon dioxide (CO2), make up more than 80 percent of the greenhouse gases emitted in the United States (EPA, 2019). The burning of fossil fuels releases CO2 and other greenhouse gases. These carbon emissions raise global temperatures by trapping solar energy in the atmosphere. This alters water supplies and weather patterns, changes the growing season for food crops, and threatens coastal communities with increasing sea levels (EPA, 2016).

The amount of CO2 emitted per year A (in tons) for a vehicle that averages x miles per gallon of gas, can be approximated by the function .

  1. Interpret the interval [20, 25] in the context of the question and then calculate and interpret the average rate of change.
  2. Determine the average rate of change of the amount of CO2 emitted in a year over the interval [35, 40], and interpret its meaning.

 

  1. Provide an interpretation of the difference between the values found in parts a) and b) and state the implications in the context of vehicle emissions of CO2. Be sure to discuss what the difference is between the average miles per gallon of 20-25 versus 35-40. Is one more ideal? Provide an explanation of your reasoning. (Department of Energy, 2019)

 

 

  • Carbon Dioxide Change

As humans continue to burn fossil fuels, the amount of CO2 in the atmosphere increases.  Scientists measure atmospheric CO2 in parts per million (ppm), which means the number of CO2 molecules for every one million molecules of other atmospheric gases such as oxygen and nitrogen.  Scientists have been tracking the amount of CO2 in the atmosphere at the Mauna Loa Observatory in Hawaii since 1958.

The table below shows the CO2 measurements recorded for the years 1959-2018.

Year Mean Year Mean Year Mean Year Mean Year Mean
1959 315.97 1972 327.45 1985 346.12 1998 366.70 2011 391.65
1960 316.91 1973 329.68 1986 347.42 1999 368.38 2012 393.85
1961 317.64 1974 330.18 1987 349.19 2000 369.55 2013 396.52
1962 318.45 1975 331.11 1988 351.57 2001 371.14 2014 398.65
1963 318.99 1976 332.04 1989 353.12 2002 373.28 2015 400.83
1964 319.62 1977 333.83 1990 354.39 2003 375.80 2016 404.24
1965 320.04 1978 335.40 1991 355.61 2004 377.52 2017 406.55
1966 321.38 1979 336.84 1992 356.45 2005 379.80 2018 408.52
1967 322.16 1980 338.75 1993 357.10 2006 381.90    
1968 323.04 1981 340.11 1994 358.83 2007 383.79    
1969 324.62 1982 341.45 1995 360.82 2008 385.60    
1970 325.68 1983 343.05 1996 362.61 2009 387.43    
1971 326.32 1984 344.65 1997 363.73 2010 389.90    

(Source: U.S. Department of Commerce/National Oceanic & Atmospheric Administration. https://www.esrl.noaa.gov/gmd/ccgg/trends/data.html )

  1. Use these data to make a summary table of the mean CO2 level in the atmosphere as measured at the Mauna Loa Observatory for the years 1960, 1965, 1970, 1975, …, 2015.

 

  1. Define the number of years that have passed after 1960 as the predictor variable x, and the mean CO2 measurement for a particular year as y. Create a linear model for the mean CO2 level in the atmosphere, y = mx + b, using the data points for 1960 and 2015 (round the slope and y-intercept values to three decimal places).  Use Desmos or Excel (directions are included in this assignment) to sketch a scatter plot of the data in your summary table and also to graph the linear model over this plot.  Comment on how well the linear model fits the data.

 

  1. Looking at your scatter plot, choose two years that you feel may provide a better linear model than the line created in part b). Use the two points you selected to calculate a new linear model and use Desmos to plot this line as well.  Provide this linear model and state the slope and y-intercept, again, rounded to three decimal places.

 

  1. Use the linear model generated in part c) to predict the mean CO2 level for each of the years 2010 and 2015, separately. Compare the predicted values from your model to the recorded measured values for these years. What conclusions can you reach based on this comparison?

 

  1. Again, using the linear model generated in part c), determine in which year the mean level of CO2 in the atmosphere would exceed 420 parts per million.

 

  • Sea-Level Rise

The Arctic ice cap is a large sheet of sea ice that contains an estimated 680,000 cubic miles of water.  If the global climate were to warm significantly as a result of the greenhouse effect or other climactic change, this ice cap would start to melt (NASA, n.d.).  More than 200 million people currently live on land that is less than 3 feet above sea level.  There are several large cities in the world that have a low average elevation, including Miami, Florida (pop. 463,347) at 7 feet, Shanghai, China (pop. 24,180,000) at 13 feet, and Boston, Massachusetts (pop. 685,094) at 14 feet.  In this part of the project you are going to estimate the rise in sea level if the ice cap were to melt and determine whether this event would have a significant impact on the people living in these three cities (US Government, 2019).

  1. The surface area of a sphere is given by the expression , where is its radius.  Although the shape of the earth is not exactly spherical, it has an average radius of 3,960 miles.  Estimate the surface area of the earth to the nearest million square miles.
  2. Oceans cover approximately 71% of the total surface area of the earth. How many square miles of the earth’s surface are covered by oceans (again, rounded to the nearest million)?
  3. Approximate the potential sea-level rise if half the Arctic ice cap were to melt. This can be done by dividing the volume of water from the melted ice cap by the surface area of the earth’s oceans.  Convert your answer into feet.
  4. Discuss what your approximation of the potential sea-level rise implies for the cities of Miami, Boston, and Shanghai. What are some possible social, economic, or political impacts for these cities and the people who live there? Based on your specified impact, what could be an appropriate and ethical response?
  5. The Antarctic ice cap contains approximately 6,300,000 cubic miles of water. Approximate the potential sea-level rise if half the Antarctic ice cap were to melt, and discuss the implications for the cities of Miami, Boston, and Shanghai.

 

References:

EPA. (2016, December 20). Climate Change Impacts | US EPA. Retrieved December 13, 2019, from https://19january2017snapshot.epa.gov/climate-impacts_.html
EPA. (2019, May 13). Overview of Greenhouse Gases. Retrieved December 13, 2019, from https://www.epa.gov/ghgemissions/overview-greenhouse-gases
Department of Energy. (n.d.). Department of Energy. Retrieved December 13, 2019, from https://www.energy.gov/
United States Government. (2019, September 5). U.S. Geological Survey. Retrieved December 13, 2019, from https://www.doi.gov/hurricanesandy/usgs
NASA. (n.d.). Arctic Sea Ice Minimum | NASA Global Climate Change. Retrieved December 13, 2019, from https://climate.nasa.gov/vital-signs/arctic-sea-ice/