Math activity

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Cubic Regression STEM Project Week #7

In the first activity we looked at a set of data that was modeled by a line (a linear regression). In the second and third activities we looked at data that was modeled by a quadratic (a quadratic regression). In the last activity we also looked more in depth at characteristics of a quadratic. In this activity we are going to model a third degree polynomial that is also called a cubic. Chemical Reaction Kinetics The field of chemical reaction kinetics (CRK) seeks to describe and predict the rates of chemical reactions. It has broad application to every industry involving chemical reactions, from oil production to microelectronics. CRK seeks to find a formula or model that describes how the conditions of a reaction affect its rate. For example, increasing the temperature of a system almost always increases the reaction rate. Two other important variables that affect the rate of a reaction are the concentration of the reactants (what goes into the reaction) and the concentration of the products (what comes out of the reaction).

Terms Chemical Concentration = amount of a chemical species per volume Chemical Species = a chemical element (such as an atom or molecule) Reaction Rate = the speed at which a reaction takes place Units L = Liter Mole = the unit for the amount of a substance = 6.022X10^(23) Sec = second

The Data Set A chemist is studying the rate of a certain decomposition reaction. A decomposition reaction is one where one chemical species decomposes into two or more new chemical species, such as the reaction below. A  B + C (A decomposes into B and C) To characterize the rate of the reaction, the chemist ran a series of experiments collecting rate data for the reaction at various concentrations of species A. The results of her experiments are shown below. Use her data to answer the questions.

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Concentration of

Species A Rate

Mole/L M/(L * sec)

0 Mole/L 5.0 M/(L * sec)

0.5 Mole/L 8.0 M/(L * sec)

1 Mole/L 27.4 M/(L *

sec)

1.5 Mole/L 55.0 M/(L *

sec)

2 Mole/L 82.6 M/(L *

sec)

2.5 Mole/L 102.0 M/(L *

sec)

3 Mole/L 105.0 M/(L *

sec)

Working With the Data Set First, find a window for your graph. Once you have found the window, label it on the grid below. Now plot your scatterplot. (Remember the vertical axis (y-axis) is the reaction rate r, and the horizontal axis (x-axis) is the concentration A).

Looking at your sketch, what type of regression best fits your data? (linear, quadratic, cubic, exponential) Explain.

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If you plotted the points correctly you should see that a cubic fits the data best. Use the directions in the Regression Instructions document to find and graph the cubic regression on your calculator. Reconcile any differences between your sketch and the graph on your calculator. Discuss the similarities between your graphs. What is the cubic equation that best fits this data? (write your answer using A and r) When you graph a cubic equation in your calculator you will notice that it does not take into consideration restrictions on your domain because of the application. What is the domain of your cubic function NOT taking into consideration the application? What is the domain of your function considering the application? (Think about what the domain represents.) What is the range of your function NOT taking into consideration the application? What is the range considering the application? (Think about what the range represents.) Do you think a linear regression or a quadratic regression might fit the data better? Explain. Questions About the Data

1. Based on the data trend, what will happen to the rate if the concentration is increased beyond 3 mole/L?

2. What interval(s) is the graph of the model increasing?

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3. What interval(s) is the graph of the model decreasing?

4. At approximately what concentration(s) is the rate increasing the slowest?

5. At approximately what concentration(s) is the rate increasing the fastest?

6. What is the approximate reaction rate when the concentration is 0.75 Mole/L? Discuss how you would find this value both algebraically and graphically.

7. What is the approximate concentration when the reaction rate is 40.0 M/(L * sec)? Discuss how you would find this value both algebraically and graphically.

Exercises:

1. For the following set of data, graph the data both by hand and on your calculator. Make sure to choose and justify a window. Discuss the domain and range for our situation. Find the cubic equation that best models the data. Discuss why a cubic equation fits best. Then discuss on which intervals the model is increasing and what intervals the model is decreasing.

Concentration

of Species A Rate

Mole/L M/(L * sec)

0 Mole/L 11.5 M/(L * sec)

0.5 Mole/L 18.4 M/(L * sec)

1 Mole/L 63.0 M/(L * sec)

1.5 Mole/L 126.5 M/(L * sec)

2 Mole/L 190.0 M/(L * sec)

2.5 Mole/L 234.6 M/(L * sec)

3 Mole/L 241.5 M/(L * sec)

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2. Discuss any similarities and any differences between the data set and graph in the example and the data set and graph in Exercise 1.

Connections: Relating previously learned skills to a new skill: We plan to provide at least one problem on each test which expands previously learned skills to a new application. Today we will be factoring higher degree polynomials and exploring the relationship between the polynomial factors, the zeros on the graph and function values. This week you are going to use the remainder theorem. When you use the remainder theorem you are finding the remainder when a polynomial is divided by a binomial. If the remainder is zero, there are a few things that we know about our polynomial, its factored form and its graph.

Let’s start with the polynomial   242178 234  xxxxxf . Find  2f .

On your calculator and on the grid below, graph   242178 234  xxxxxf .

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Make a connection between the graph of  xf and the value found for  2f . Looking at the graph of  xf , what other values for x will give a remainder of zero? Now, we will factor  xf using the conclusions reached above. What factor corresponds to the x- intercept of 2? Use the graph of  xf to factor  xf . Now, use the graph of  xf , to determine when   0xf . Indicate this solution on your graph.

Now let’s look at the equation 8x 3 + 24 = x

4 +17x

2 + 2x; list two ways that you can solve this

equation graphically. Then choose the best method and solve it graphically. How would you solve the equation algebraically? Solve the equation algebraically and check your solution with your graphical solution. More examples: Solve the following equations both graphically and algebraically.

1. Graph and solve 23

21013 xxx  . Then factor the expression 10132 23

 xxx .

2. Graph and solve 10x 3 + 50x = x

4 + 35x

2 + 24. Then factor the expression

24503510 234

 xxxx .