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Math 4b Class Project: Modeling Real World Relationships to Make Predictions Purpose: To use the concepts developed in Precalculus to model an actual real world relationship and make a prediction

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Math 4b Class Project: Modeling Real World Relationships to Make Predictions

Purpose: To use the concepts developed in Precalculus to model an actual real world relationship and make a prediction. In this case we will model global warming

The data: The following table summarizes the average yearly temperature (F°) and carbon dioxide emissions in parts per million (ppm) measured at Mauna Loa, Hawaii

Year	1960	1965	1970	1975	1980	1985	1990	1995	2000	2005
Temperature	44.45	43.29	43.61	43.35	46.66	45.71	45.53	47.53	45.86	46.23
CO2 Emissions	316.9	320.0	325.7	331.1	338.7	345.9	354.2	360.6	369.4	379.7

 

Defining our variables: t = years after 1960, T = Temperature, and C = CO₂ emissions.

I. Describe the Relationship with a Quadratic Functions in vertex form

We will use the data from the years 1960 and 1990 for our models.

That is, fort = 0, T = 44.45 and C = 316.9 while for t = 30, T = 45.53 and C = 354.2

1) Modeling Temperature

a) Use the data from 1960 and 1990 to find a quadratic function that models the Temperature. Use the first data point as the vertex.

b) Use your quadratic function to predict the Temperature in 2005. Compare your prediction with the actual Temperature in 2005.

c) On graph paper, plot the entire set of temperature data from 1960 through 2005. On the same axis, plot your function from part (a) with at least 4 actual points on the graph to give it some adequate scale. Discuss the similarities and differences of the graph and the data. How well does your function approximate the actual data compared to the linear function you used in part 1?

2) Modeling CO₂ emissions

a) Use the data from 1960 and 1990 to find a quadratic function that models the CO₂ emissions. Use the first data point as the vertex.

b) Use your quadratic function to predict the CO₂ emissions in 2005. Compare your prediction with the actual CO₂ emissions in 2005.

c) On graph paper, plot the entire set of CO₂ emissions data from 1960 through 2005. On the same axis, plot your function from part (a) with at least 4 actual points on the graph to give it some adequate scale. Discuss the similarities and differences of the graph and the data. How well does your function approximate the actual data compared to the linear function you used in part 1?

ll. Describe the Relationship with a Quadratic Functions in standard form

We will use the data from the years 1960, 1990, 2005, for our models.

That is, fort = 0, T= 44.45 and C = 316.9 and for t = 30, T = 45.53 and C = 354.2, etc.

3) Modeling Temperature

 

a) Use the data from 1960, 1990, and 2005 to find a quadratic function T = at² + bt + c that models the Temperature in terms of the years since 1960.

b) Use your quadratic function to predict the Temperature in 2020.

c) On graph paper, plot the entire set of temperature data from 1960 through 2020. On the same axis, plot your function from part (a) with at least 4 actual points on the graph to give it some adequate scale. Discuss the similarities and differences of the graph and the data. How well does your function approximate the actual data compared to the linear function you used in part 1 and the quadratic function in vertex form in Part 2 #1?

4) Modeling CO₂ emissions

d) Use the data from 1960, 1990, and 2005 to find a quadratic function C = at² + bt + c that models the CO₂ emissions in terms of the years since 1960. (Hint: Plug in each data point to get a system of equations. Solve the system to find a, b, and c.)

e) Use your quadratic function to predict the CO₂ emissions in 2020.

f) On graph paper, plot the entire set of CO₂ emissions data from 1960 through 2005. On the same axis, plot your function from part (a) with at least 4 actual points on the graph to give it some adequate scale. Discuss the similarities and differences of the graph and the data. How well does your function approximate the actual data compared to the linear function you used in part 1 and the quadratic function in vertex form in Part 2 #1?