Jeff Levine and Melissa Horak

Team 3

Jeff Levine and Melissa Horak

Optimization Problem

Student Inquiry-based Activity/Lesson

Math Technology

Spring 2008

April 10, 2008

(source for picture:

Optimization

What is optimization?

An optimization problem is determining a function to maximize or minimize. For example, in a manufacturing company, the business may want to maximize the profit or minimize the cost of the product. Or, if a company is designing an automobile panel, they might want to maximize the strength of the product. In these problems, there is a set of variables, or unknowns, that affect the value of the function. For the manufacturing company example, these variables may include the amount of different resources used or the amount of time spent on each activity. Lastly, a set of constraints allow these variables to take on specific values, but may exclude other values. For the manufacturing example, the company would not spend a negative amount of time on an activity, so there would be a constraint to make the time variables non-negative.

*** So, in conclusion, the optimization problem is a problem that finds the values of the variables, or unknowns, that minimize of maximize the objective function while satisfying the constraints.

Activity 1:Using tiles, pencil, ruler, and construction paper.

A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are each 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

On your desk, you are given a piece of 18x24 inch piece of construction paper, 24 sq. in. green tiles, approximately 100 sq. in. black tiles, a pencil, and a ruler.

Arrange the green tiles into rectangles. Then, draw the rectangles on the construction paper given.

Now, extend the margins of each of your rectangles by one inch on the left and right hand sides. Next, extend the top and bottom or the rectangles by 1.5 inches.

Now, fill these new rectangles with the black tiles.

How many tiles fit in the new rectangles?

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Compare the ratio of the sides of each old rectangle to its new rectangle. What does this activity show? Explain.

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Activity 2:Using Algebra to solve the problem.

We are going to take a look at solving this problem, using algebraic methods and the TI-nSpirecalculator.

From the information in the problem above, create an equation for the area of the for the least amount of paper used. Remember, the total area of this print is 24 square inches and the margins need to be included in the equation.

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Next, we need to make an equation, from #1, but we need to substitute for one of the variables. We know that the area of the print is xy = 24. If we set the “x” variable as the independent variable, we must solve for y, with respect to “x” in the equation. What is the new equation going to be, after we replace y, with x?

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Now that we have an equation with only one variable, we need to foil-factor the equation, so we can use this information in the calculator. What is the new equation that we will be working with?

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Now, we are going to use the TI-nSpire. Go to the home page and select the graphs and geometry section.

At the bottom of the screen, there is a small box. Put our newly created equation into this box and hit enter.

Does the entire graph show with the given window dimensions? If not, determine what dimensions would better represent the equation and write them here:

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After determining a the dimensions, we see a graph that shows positive and negative x values. Do we need both positive and negative values? Discuss here:

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Since we will never have negative dimensions, we need to only look at the non-negative values of x. So, re-do your dimensions, so that we are only seeing the values that are possible for the problem.

Make some general comments about your graphs here: (What does the graph represent? How do we determine what the value of x should be, etc)

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Next, we are going to create a table to see where the minimum area will occur. Click menu and choose “View.” Scroll to “Add Function Table” and click “enter.”

The table will appear on the screen. To go back and forth between the table and the graph, press “Ctrl” and “tab.”

Scroll through the table values for y. What do you see? Where is it decreasing and increasing?

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We need to use the numerical-zoom in strategy to determine what the x-value is for the problem. This x-value should be where the minimum area occurs.

To do this, click “menu” and choose “Function Table.” Then, choose the sub-topic “Edit Function Table Setting.”
Under the table start section, set at 5.0. Then, use the table step at 0.1. Why are we doing this?

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After changing the table settings, view the new table. What do you see?

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After zooming in the table, what is the x-value for the area for the inches of print?

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EXTENSION:

What are some properties and definitions we discussed in this activity?

After determining the x-value is six for the area of the inches of print, what are the dimensions of the page? Remember to use the correct equation to solve this. Show your work here:

Activity 3:Using geometry to solve the problem.

The TI-nSpire has some unique features that will allow you to actually draw the rectangular text box, move the dimensions of the rectangular, capture the data, and analyze the data.

1.Open up the graphs and geometry function of the calculator. Then, hide the axes as well as the entry line. You are left with a “drawing” board.

2.Using the drawing tools, draw a rectangle. Then, move one of the points around so that the area of the rectangle is 15 sq. units. (In the example, the problem is in inches, but on the calculator, often times the units are in centimeters. That is OK.)

3.Now re-scale appropriately so that the area of the rectangle is 24 sq. units, accurate to at least 3 decimal places.

4.Now, without erasing any of the data you just created, create another window, but this time, a window with a spreadsheet. Then, label appropriately the first 5 columns as follows:

1st column – the base of the rectangle you created

2nd column – the height of the rectangle you created

3rd column – the width of the piece of paper; you will need to create a formula to take the data from column 1 and add-on the appropriate measure for the margins.

Write down the formula used in column 3.

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4th column – the length (top-to-bottom) of the piece of paper; you will need to create a formula to take the data from column 2 and add-on the appropriate measure for the margins.

Write down the formula used in column 4.

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5th column – the area of the piece of paper; you will need to crate a formula to take the data from columns 3 and 4 and multiply them together.

Write down the formula used in column 4.

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*** Take your time with this step and the next step. MAKE SURE that you label your columns appropriately, you use correct formulas, and you are grabbing data from the correct columns when setting us the formulas.***

Now set-up the templates so that as you adjust the size of the rectangle, the base is captured in column 1 of your spreadsheet and the height is captured in column 2 of your spreadsheet. After you do this, check to be sure as you move a corner of the rectangle, data is being captured into your spreadsheet, and columns 3-5 are being calculated correctly.

6.Summarize your findings. Why is the area of the piece of paper not the least when the piece of paper is a square? When is the area the least? Why? Discuss any other peculiar finding of anything else of interest that you find.

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Extension Activity 3: Regression analysis with spreadsheet data.

Now that you have a spreadsheet full of interesting data, let’s do a “regression analysis.” Essentially, this is a process of taking data and find an equation to best fit the data.

Before we do that, let’s scatter plot the data to give us a better idea what type of equation will fit the data. Set up a new screen graphing screen. Then, change the function type to “scatter plot.” For your x-value, select either the width or the length of the piece of paper. For the y-value, select the other.

You will notice that what appears is part of a graph. This is actually just the points plotted from columns 3 and 4 of your spreadsheet. To see this more clearly, either zoom-in manually with the window settings, or zoom-in using the “zoom-box” feature. Now, you can tell that what you are looking at is a bunch of points plotted on the graph.

There are many types of regression equations. To help you decide which one, write down the following:

What is the equation to calculate a side of the rectangular box of print given another side?

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Re-write this equation so that “x” is no longer in the denominator. In other words, what power is “x” raised to?

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Open up a new calculator window and start searching through the possible regression equations (hit “menu,” then scroll down to “7: Statistics,” then move to the right and highlight “1: Stat Calculations,” then hit the right arrow and all the possible choices appear.) Which regression equation do we want to run? Why?

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Run this regression equation. For the “x” and “y” values, enter the same “titles” you used in your scatter plot. What is your regression equation?

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This equation is stored as a function. Graph the equation. Talk about what window settings make sense to view the part of this regression equation that makes sense for this problem.

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Now, go around to a few other students and talk about your results. Some students may have a different regression equation. Write down that one here:

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Discuss why there are 2 different types of regression equations for the same problem (some with the exponent >-1 and some with the exponent <-1. Also, discuss the possible reasons for the coefficients and exponents when compared to each other in each equation.

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Why are there many different regression equations with r=~.99 for the same problem?

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If you set one of each type of regression equation equal to each other (one with the exponent >-1 and one with the exponent <-1), what is the significance of the answer.

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Activity 4:Discussing the problem using a calculus method.

Now that we have discussed the problem using TI-nSpire to find a function that models the problem situation and also to draw a picture using the dynamic geometry software, we are going to discuss how to look at this problem through calculus.

1.From the information in the problem above, create an equation for the area of the for the least amount of paper used. Remember, the total area of this print is 24 square inches and the margins need to be included in the equation.

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2.Next, we need to make an equation, from #1, but we need to substitute for one of the variables. We know that the area of the print is xy = 24. If we set the “x” variable as the independent variable, we must solve for y, with respect to “x” in the equation. What is the new equation going to be, after we replace y, with x?

______

Now that we have an equation with only one variable, we need to foil-factor the equation, so we can use this information in the calculator. What is the new equation that we will be working with?

______

Now we are going to use our TI-nSpire calculators. Press the catalog button to begin the calculus portion of solving this problem.

After pressing this button, you will see five different tabs to choose from. Press the number 2 and use the down arrow to scroll down to the word Calculus.

Press the right arrow button and this expands the Calculus section. It will divide into multiple subsections.

The first subsection is the word derivative. Make sure your selection is on the word derivative and press enter.

An image like this will appear on your screen. What does this image represent when discussing a function? What is the definition of a derivative, when discussing a function?

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Now, in the box next to the “d” in the denominator, put “x”. Why are we putting the letter “x” into this part of the equation?

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Now, the box inside the parenthesis is where we are going to put in our function equation. Type in 72/x + 2x+ 30 and then press enter. What is the result? What does this new equation represent?

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After determining the derivative of the function, we need to solve for the possible x values. Click menu and choose Algebra. In the sub-topics of Algebra, choose solve.

After selecting solve, it will appear in the screen. In the parenthesis next to solve, put in the derivative of the equation and set it equal to zero. But, we also need to add a comma and an “x” after the equation and close the parenthesis. Hit enter.

What does the calculator show as your possible answers? What does this mean? Are there any values that may not be included, even though they are possible answers for the equation? Explain in detail.

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Now we are going to graph the equation and the derivative to see other options for the results. Go to the home page and click on the graphs and geometry.

In f1(x), put in your equation and in f2(x), put in the derivative. Press enter.

Both equations should appear on the screen. What are the best dimensions for each of the equations that will show the complete graph of the values we are looking for. (The dimensions should be different for each equation)

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Which of the graphs gives a better representation of the data we need? Why?

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What are the values of the x and y for the print and for the least amount of paper used?

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Show work here:

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