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General Education Common Graded Assignment: MATH 163 Spring 2020
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:
    Written and Oral Communication articulate a solution to mathematical problems (CCO20),
    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),
    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).

In addition to the above general education objectives, this assignment assesses students understanding and application of the following skills and knowledge specific to applications of linear equations and modeling:
    Modeling data with linear regression function

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 Earths 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 all parts of each scenario 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: 
    Minimum 7 pages typed in APA format including appropriate title page, font, and margins. Directions can be found on the CCBC library page. The title page and Reference page are part of the document but do not count towards the page total.
    Cut and paste all technology displays and outputs, as needed, directly into the final document.
    If needed, use APA style format and documentation for parenthetical citations and a References page.
    Utilize academically appropriate resources such as those found in library databases and elsewhere on the internet

GRADING:
    This assignment will account for 10% of the total course grade.
    See attached rubric for details about how your essay will be graded

SUBMISSION GUIDELINES:
    [INSERT DUE DATE] This date must be in the last 1/3rd of the Assessment semester
    For paper submissions, please submit TWO copies. 
One copy should include your student ID, course number and section and omit student and faculty names.
The other will be evaluated within your course and should include the student name.
    Electronic submissions should be made with the students ID number (900 or 901#) as the file name through Blackboard. Student and instructor names should not appear on electronic submissions.

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 A(x)=0.0089x^2-0.815x+22.3.
    Determine the average rate of change of the amount of CO2 emitted in a year over the interval [20, 25], and interpret its meaning.

    Determine the average rate of change of the amount of CO2 emitted in a year over the interval [35, 40], and interpret its meaning.

    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.
(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 )
    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.

    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 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.

    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.

    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?

    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).
    The surface area of a sphere is given by the expression A=4r^2, where r 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.

    Oceans cover approximately 71% of the total surface area of the earth.  How many square miles of the earths surface are covered by oceans (again, rounded to the nearest million)?

    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 earths oceans.  Convert your answer into feet.

    Discuss what your approximation of the potential sea-level rise implies for the cities of Miami, Boston, and Shanghai.

    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.

    Air Pollution Reduction –  Cost-Benefit Analysis
Coal has long been a reliable source of American energy, but it comes with tremendous costs because it is very dirty.  When coal is burned it releases a number of airborne toxins and pollutants, including mercury, lead, sulfur dioxide, nitrogen oxides, particulates, and various other heavy metals which can have harmful environmental impacts in addition to CO2.
The function below relates the cost C (in $1000) to remove x percent of the air pollutants for a hypothetical power company which burns coal to generate electricity.  A function such as this is called a cost-benefit function because it relates a cost (the price of implementing practices to remove the pollution) and a benefit (the removal of the air pollutants from the energy-generating process).
C(x)=540x/(100-x)  for 0x<100
    Use the above function to show that the cost of removing 40% of the air pollutants would be $360,000.  Then compute the cost (in dollars) for the company to remove 50%, 55%, 60%, 65%, 70%, 75%, 80%, 85%, 90%, and 95% of the air pollutants.  Organize your results in a table and make a scatterplot of the points using Desmos.  What trend do you observe?

    What would be the increased cost (in dollars) for the company if they were to add processes that would increase the amount of pollutants removed from 50% to 60%, from 60% to 70%, from 70% to 80%, and from 80% to 90%?  Comment on any trend that may be noticed.

    According to the cost-benefit function, would it be possible for the company to remove 100% of the air pollutants?  Explain why this does or does not make sense.

    The domain of C(x) is restricted to 0x<100.  Explain why this makes sense in the context of this model.

    If the company decides they can reasonably budget $2 million for pollution control, what percentage of air pollutants can be removed (to the nearest tenth of a percent)?

    Sketch the graph of C(x) (for  0x<100) using Desmos.  What is the range of C(x)?  Explain why the range makes sense in the context of the problem.

Instructions 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.
    To create a new graph, Go to Desmos.com and 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.
    Finding an equation that best fits your data in Desmos
    Go to Desmos.com and choose Start Graphing.
    Click the plus sign in the upper left and choose Add Item > table
    Type your data in the table.
    Click on the wrench in the upper right to change the graph settings.
    Modify your x, and y values to reflect your data. 
    Adjust the values of the sliders until the graph of the equation most closely fits your data points.
    On the DESMOS Calculator, type y=mx+b 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.
Helpful videos:
https://learn.desmos.com/tables
https://learn.desmos.com/graphing
https://www.youtube.com/watch?v=4NTf551hHQE

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/

 

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