Math project

Applications Project

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 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 theissuesclimate 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 thispre-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.

Math 163 Applied Project

1. Carbon Dioxide Emissions

Carbon emissions contribute to climate change, which hasserious 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 releasesCO₂ 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 averagesx miles per gallon of gas, can be approximated by the function.

1. 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 CO₂in the atmosphere increases.  Scientists measure atmospheric CO₂in parts per million (ppm), which means the number of CO₂molecules for every one million molecules of other atmospheric gases such as oxygen and nitrogen.  Scientists have been tracking the amount of CO₂in the atmosphere at the Mauna Loa Observatory in Hawaii since 1958.

The table below shows theCO₂ 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 CO₂level in the atmosphere as measuredat 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 CO₂level 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 CO₂level for each of the years2010 and 2015, separately. Compare the predicted valuesfrom 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).

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.

• 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)?

• Approximate the potential sea-level rise if half theArctic 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.

• 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 theAntarctic 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).

1. 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  is restricted to .  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  (for  ) using Desmos.  What is the range of ?  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.

1. 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.
2. Finding an equation that best fits your data in Desmos
3. Go to Desmos.com and choose Start Graphing.
4. Click the plus sign in the upper left and choose Add Item>table
5. Type your data in the table.
6. Click on the wrench in the upper right to change the graph settings.
7. Modify your x, and y values to reflect your data. 
8. Adjust the values of the sliders until the graph of the equation most closely fits your data points.
9. 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.

Helpful videos:

https://learn.desmos.com/tables

https://learn.desmos.com/graphing

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/