In this set of exercises, you will use polynomials to study...

In this set of exercises, you will use polynomials to study real-world problems. Sulfur dioxide $\left(\mathrm{SO}_{2}\right)$ is emitted by power-gencrating plants and is one of the primary sources of acid rain. The following table gives the total annual $\mathrm{SO}_{2}$ emissions from the 263 highest-emitting sources for selected years. (Source: Environmental Protection Agency) TABLE CANT COPY (a) Let $t$ denote the number of years since $1980 .$ Make a scatter plot of sulfur dioxide emissions versus $t$ (b) Find an expression for the cubic curve of best fit for this data. (c) Plot the cubic model for the years $1980-2005$ Remember that for the years $2001-2005,$ the curve gives only a projection. (d) Forecast the amount of $\mathrm{SO}_{2}$ emissions for the year 2005 using the cubic function from part (b). (c) Do you think the projection found in part (d) is attainable? Why or why not? (f) The Clean Air Act was passed in $1990,$ in part to implement measures to reduce the amount of sulfur dioxide emissions. According to the model presented here, have these measures been successful? Explain.

In this set of exercises, you will use polynomials to study real-world problems.
Sulfur dioxide $\left(\mathrm{SO}_{2}\right)$ is emitted by power-gencrating plants and is one of the primary sources of acid rain. The following table gives the total annual $\mathrm{SO}_{2}$ emissions from the 263 highest-emitting sources for selected years. (Source: Environmental Protection Agency)
TABLE CANT COPY
(a) Let $t$ denote the number of years since $1980 .$ Make a scatter plot of sulfur dioxide emissions versus $t$
(b) Find an expression for the cubic curve of best fit for this data.
(c) Plot the cubic model for the years $1980-2005$ Remember that for the years $2001-2005,$ the curve gives only a projection.
(d) Forecast the amount of $\mathrm{SO}_{2}$ emissions for the year 2005 using the cubic function from part (b).
(c) Do you think the projection found in part (d) is attainable? Why or why not?
(f) The Clean Air Act was passed in $1990,$ in part to implement measures to reduce the amount of sulfur dioxide emissions. According to the model presented here, have these measures been successful? Explain.

Key Concepts

Forecasting

Forecasting involves using mathematical models, such as fitted polynomial curves, to predict future values based on existing data trends. It is a common technique in various fields for planning and decision-making, but the reliability of the forecast depends on the validity of the assumptions underlying the model.

Extrapolation

Extrapolation is the process of estimating the value of a variable outside the range of the observed data by extending a model. While it can provide predictions beyond the available data, extrapolation carries risks because the model's behavior may not accurately represent the true relationship outside the observed interval.

Model Evaluation

Model evaluation involves assessing the accuracy and reliability of a mathematical or statistical model in representing real-world data. This includes verifying whether the fitted model aligns with known events or interventions and understanding its limitations, especially when predicting outcomes beyond the range of the original data.

Polynomial Regression

Polynomial regression is a form of regression analysis that models the relationship between a dependent variable and one or more independent variables as an nth-degree polynomial. It is used to fit a curve to observational data and is particularly useful when data exhibit non-linear behavior that simpler linear models cannot capture.

Scatter Plot

A scatter plot is a graphical representation of paired numerical data, where each value in the dataset is represented as a point on a Cartesian plane. This type of plot helps to visually inspect the relationship between two quantitative variables and can reveal trends, clusters, or outliers in the data.

Cubic Functions

Cubic functions are a specific type of polynomial function of degree three. They have the general form ax³ + bx² + cx + d and are capable of modeling more complex relationships than linear or quadratic functions. In data modeling, they are often used when the relationship between variables follows an s-shaped or inflection pattern.

Transcript

00:01 We are using polynomials in real life.

00:04 And so we are studying, we are looking at the sulfur dioxide by higher generating plants.

00:13 And we want to know that after 1980, we have a set of data from 1980 to 2000.

00:24 And yes so from 1980 to 2000 so from here from here this is this is 1980 so let's say let's say 1980 is where is where t is equal to zero so that's where t is equal to zero and we have that 2000 is where t is equal to 20.

01:08 And so we can see that on our t -axis here.

01:13 And this is, let's say this is our y.

01:19 That is our y.

01:21 And so we can see here that i've plotted, i've plotted the data in google sheets, and i've created a cubic, cubic best fit function.

01:36 And so the the function follows this sort of curve here that tries to get all the points within our set of data.

01:49 And so the first thing we were asked is to graph this.

01:53 Well, i have this scatter plot, this scatter plot as well as this cubic function.

01:59 And so we are asked to find the cubic function of best fit.

02:06 And so we've done that.

02:07 We've done that on google sheets by plotting this data, plotting the given data.

02:15 And then we want to know, okay, what does this graph look like if we were to do, if we were to do from 1980 to 2005? and 2005 would be, we would say, t is equal to 25.

02:32 So we want this graph.

02:33 We want this function from zero to 25.

02:37 And so i've graphed that here.

02:39 So we have it from zero to 25.

02:43 And so next we are asked to give a projection for 2005.

02:53 And so what 2005 is, is it's right here.

03:00 Let's say 2005 is right here.

03:05 And so if this value, if this is two, i want to say, say it's right underneath, right below.

03:15 So let's say, let's say our value is 1 .7.

03:22 So then we're asked, and this would be in millions of tons, so millions of tons, millions of tons...

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