S7 - MODELING WITH THE TI-84

Transcription

1 Summer 2006 I2T2 Probability & Statistics Page 137 S7 - MODELING WITH THE TI-84 Investigating Elasticity Objective: To have students test their understanding of the equation of a line to fit data from an experiment. Concepts/Skills: Students perform an experiment, collect data, graph the data and fit an equation of a line to the data. Needs: Paper cup String to hang cup Rubber band or slinky Stand or yardstick supported across two chairs About 50 pennies, either pre-1982 or post Ruler or measuring tape Getting started: 1. Prepare a cup by punching holes and running string across the top. Attach either a rubber band or a slinky to the string. If using a rubber band, attach a paper clip to the top. 2. Hang the cup from a ring stand or from a yardstick so that it is hanging freely. The textbook suggests taping the paper clip to the edge of a table so that it hangs over the edge. This is fine as long as neither the cup nor the rubber band is hitting the table. 3. If using the edge of the table, you can tape a ruler or measuring tape to the edge near the cup. Have 0 at the top of the rubber band. Otherwise, use a ruler, yardstick or measuring tape to measure the stretch of the rubber band or slinky. 4. Record the distance from the top of the rubber band or slinky to the bottom of the cup. This is the initial distance. 5. Add 10 pennies to the cup. Record the number of pennies in the cup and the distance from the top of the rubber band or slinky to the bottom of the cup. 6. Repeat Step 5 several more times until you have 50 pennies in the cup. Each time, record the number of pennies in the cup and the distance from the top of the rubber band or slinky to the bottom of the cup. pennies distance

2 Summer 2006 I2T2 Probability & Statistics Page 138 Investigating the data: 1. Make a scatter plot of the data you have collected. Describe any patterns you see. 2. Find an equation for a best-fitting line. You may want to use a graphing calculator or computer. 3. Explain what the y-intercept and the slope mean in terms of your data. 4. Describe a reasonable domain and range of your equation. Write inequalities to represent both. Making conjectures: 5. Make a conjecture about what the distance from the top of the rubber band or slinky to the bottom of the cup would be if you placed 100 pennies in the cup. 6. Test your conjecture. What relationship do you see between the number of pennies used and the length of the stretch? (Actually this is not a linear relationship as suggested by the book. After a certain point, the rubber band will not continue to stretch at the same rate.) 7. Make a conjecture about how the length of the rubber band or slinky might affect the total distances. Would the equations be the same or different? 8. How would your graph be different if you measured from the floor up to the bottom of the rubber band or slinky? (Using the CBR, you are measuring from the floor to the bottom of the cup. Note: Using Hooke s Law to Explore Linear Functions by Chris McGlone and Gary Nieberle, Mathematics Teacher, May 2000, is a similar activity. Using the Graphing Calculator: 1. Make a scatter plot of the data you have collected. a. Using the graphing calculator, press STAT EDIT, enter. Clear the lists L1 and L2. Enter the values for the number of pennies in L1. Enter the corresponding distances in L2. b. Go to STAT PLOT by pressing 2nd Y= Press ENTER to go to Plot 1. Press ENTER to turn Plot 1 on. Select the scatter plot (first choice). For Xlist enter L1, for Ylist enter L2. Choose one of the first two marks. c. Press ZOOM then choose 9:ZoomStat. d. Describe any patterns you see. 2. Find an equation for a best-fitting line. a. Go to CATALOG (2nd 0). Press D (ALPHA x 1 ). Arrow down to DiagnosticOn. Press ENTER twice, the word Done should be on the screen. b. Go to STAT CALC. Arrow down to 4:LinReg(ax+b). Press ENTER. After LinReg(ax+b) enter L1 (2nd 1), then comma (above 7), then L2 (2nd 2). Now press VARS Y-VARS, choose 1:Function, then 1:Y1. On the main screen press ENTER. You should see values for a and b. The values for r and r 2 indicate how closely the equation fits the data. The closer r and r 2 are to 1 or 1, the better the fit. c. The equation has been pasted into Y1. Press graph to see a graph of the line. Compare the line to the plotted data.

3 Summer 2006 I2T2 Probability & Statistics Page 139 Area and Perimeter: Building a Garden Fence Adapted from Graphing Calculator Activities for Enriching Middle School Mathematics, Browning & Channell, Texas Instruments, Concepts/Skills: Geometry, measurement, dimensions, area, maximum area given a fixed perimeter. Needs: Graphing calculators, toothpicks or raffle tickets, grid paper or dot paper, student worksheet, calculator instruction sheet. Getting started: Tell students the following story: You and a friend are visiting her grandparents on their small farm. They have asked the two of you to design a small, rectangular-shaped vegetable garden along an existing wall in the backyard. They wish to surround the garden with a small fence to protect their plants from small animals. W W L To enclose the garden, you have 24 sections of 1 meter long rigid border fencing. In order to grow as many vegetables as possible, your task is to design the fence to enclose the maximum possible area. How many sections of fencing should you use along the width and the length of the garden? There are many rectangular shapes that can be formed using the 24 fencing sections and, before the digging begins, you should do some calculations. Procedure: Have students use their manipulatives (toothpicks or raffle tickets) to represent the sections of border fencing. They should fill out the first three questions on the student worksheets. Once they have answered the initial questions and filled out the table, work with students to enter the data into the lists on the calculator, graph the results, and answer the remaining questions on the Question sheet.

4 Summer 2006 I2T2 Probability & Statistics Page 140 Additional Explorations: The following explorations use the list capabilities of the graphing calculator to investigate the situations. In each case, students should produce a scatterplot of the widths and areas. Students will continue to use the 24 sections of fencing in forming the border. 1. A friend suggests that you plant your grandparent's garden at the back corner of the yard so that the existing fence can border two of the four sides of your garden. What are the dimensions of the garden with the largest possible area? Is this configuration an improvement over the original plan? Explain your reasoning. W L 2. Suppose the garden were placed at the corner of a barn so that it was positioned as shown below. The garden has an L-shaped configuration as it wraps around the corner of the bard. What dimensions would give the largest garden area? barn W W L L

5 Summer 2006 I2T2 Probability & Statistics Page 141 Student Questions for Garden Fence 1. If you were to use three sections of fencing along each width of the garden, how many sections would remain to form the length? What will be the area of this garden? Copy these values into the Table below, and then enter three more possible garden sizes into the table. Try to guess the width and length of the garden with the largest possible area. Compare your results with others in your class. Possible Dimensions of Garden Fence Width (m) Length (m) Area (m 2 ) 3 2. If you know what the width is, how can you find the length? Write an equation that shows this relationship between width and length. 3. The smallest number of fencing pieces you can use along the garden width is one. What is the largest number of pieces that you can use along the width of the garden? Explain how you know this. Use calculator instruction sheet to enter L1 and L2 in your calculator. 4. How can the values for L3 (the areas) be determined from L1 and L2? Remember that L1 stores the possible widths and L2 stores possible lengths. Enter L3 into your calculator. 5. Scroll through the values in L3. Are the values you computed earlier contained in this list? Describe any patterns you see in the data values contained in L3.

6 Summer 2006 I2T2 Probability & Statistics Page Examine the third list to find the dimensions of the rectangular garden that has the largest possible area. Complete the following sentence to provide a solution to the original question: A rectangle with a width of meters and a length of meters gives the largest possible garden of square meters. Return to instructions on Displaying a Graph. 7. When creating the scatterplot of the areas, you entered the settings shown at right for the display window. Why do you think these values were used? 8. You used TRACE to move through the data points in the scatterplot. Which point corresponds to the maximum area? What sets it apart from the other points on the plot? 9. How do any patterns that you observed in the lists show up in the scatterplot of the data?

7 Summer 2006 I2T2 Probability & Statistics Page 143 Instructions for using the Calculator for Garden Fence 1. To clear the first three lists in the calculator, press STAT 4:ClrList ENTER, and then press 2 nd [L1], then a comma, 2 nd [L2], then a comma, 2 nd [L3] ENTER. You will use L1 to store the possible widths and then calculate values for the corresponding lengths and areas. Once you calculate the values, you will store the lengths in L2 and the areas in L3. 2. Press STAT 1:Edit and press ENTER. Note: If L1, L2, and L3 are not visible, then press STAT 5:SetUpEditor ENTER. Then repeat step Enter the whole numbers from 1 to 11 (the largest possible width) into L1 on your calculator by typing each number and pressing ENTER until all widths have been entered. (See the example below.) You should have noticed in your earlier computations that the number of fencing pieces remaining for the length of the garden can be found by subtracting twice the number used for a width from the 24 fencing pieces available. We want to store in L2 the lengths that correspond to the widths in L1. 4. Press to move to the second list. Press to move to the top so that L2 is highlighted. Press Enter. 5. Enter [L1] as the definition for L2. Your display should look like the example at the right. 6. Press ENTER and the column of possible lengths should appear in L2. Answer #4 on the Question sheet. This will help you find an expression to enter for L3. 7. Press to move to the third list and then press to move to the top so that L3 is highlighted. Press Enter. 8. Enter 2 nd [L1] x 2 nd [L2] ENTER to enter the expression for L3 that you found in question 4. Answer #5 and #6 on the Question sheet.

8 Summer 2006 I2T2 Probability & Statistics Page 144 Displaying a Graph for Garden Fence A scatterplot is often used to present a visual display of the relationship between two sets of paired data like the width and area measurements. Your calculator can produce a scatterplot display of the numbers in its lists. 1. Press 2 nd [STAT PLOT] ENTER. Press Enter to turn on Plot 1. Edit the window so that yours looks like the one at the right. To make a selection, press to move the blinking cursor on top of the desired location and then press ENTER. 2. Press WINDOW and edit the numbers so that your window looks like the one shown at the right. 3. Press Y= and clear any equations in any of the lines. Press GRAPH to view a scatterplot. The horizontal axis represents the garden widths and the vertical axis shows the corresponding areas. 4. You can view the coordinates of each plotted point by pressing TRACE followed by the left and right arrow keys. Answer #7, #8, and #9 on the Question sheet.

9 Summer 2006 I2T2 Probability & Statistics Page 145 A Fence for Fido: Area & Perimeter Adapted from T3 MSM Summer Institute, Texas Instruments, Concepts/Skills: Geometry, measurement, dimensions, area, maximum area given a fixed perimeter. Needs: Graphing calculators, toothpicks or raffle tickets, grid paper or dot paper, Student question sheet. Getting started: Suppose we want to make a rectangle with a perimeter of 12. What are the possible rectangles that could be made? Procedure: Divide the students into small groups. Each group will need 12 toothpicks or raffle tickets, to represent the perimeter of a rectangle, and a sheet of grid paper. As a group, students are to arrange the toothpicks/tickets into every possible rectangular shape to represent the given perimeter of 12. Sketch each figure on the grid paper. Record the dimensions of each rectangle. Compile the data from every group into one chart (similar to the one below). Figure the area of each figure. Length Width Perimeter Area Present the problem: Tell students the following story: Fido, our mutt, keeps running away and we need to fence him in. The fencing costs $3 a foot and we have only $126 to spend. How much fencing material can we buy? Fido is a very hyperactive dog. We need to build the largest possible rectangular pen so he has plenty of room to run. What do the dimensions of the pen need to be? a. Sketch the possible rectangles on the grid paper b. In the cooperative group, compile all of the data on a chart c. As a class compile all of the data into one chart. 1. Look for patterns, compare width and length, compare dimensions with the area. 2. Discuss the maximum area of the dog pen. Are there solutions that are not whole numbers that would give us a bigger area? Have students enter the class data in a graphing calculator using lists. Instructions are basically the same as for the Garden Fence. First have them clear L1, L2, L3, L4.

10 Summer 2006 I2T2 Probability & Statistics Page 146 Discuss how they can write an expression for the length, width, perimeter, and area in terms of the length. Listed are examples and there may be other equivalent expressions that are valid. The amount of fence we can afford would be $126/$3 = 42, which would be the perimeter. Then half the perimeter is 21. They can calculate L2 using 21 length, or use (42 2*length)/2. a. L1 = length b. L2 = width (L2 = 21 c. L3 = perimeter (L3 = 2L d. L4 = area (L4=L1* Using the lists, what is the maximum area of Fido's pen? What are the dimensions of the pen? Students can view a graphical solution to the problem with a scatterplot. Instructions are basically the same as for the Garden Fence. Ask students which two lists they should use to provide a graphical solution to the problem. Have them explain why. a. x-axis (length) = L1 b. y-axis (area) = L4 Have them predict what the graph will look like based on the information in the lists. Selection of the window values also provide good information on the students' understanding of the constraints of the problem. Using TRACE compare the ordered pairs to the corresponding values in the lists. (See questions on Student Question sheet.) If the length = x, then how can you represent the area as a function of x? area = x(21-x) (See questions on Student question sheet. ) Students can also view the relationship between the length and area by using the table feature of a graphing calculator. By changing the ΔTbl, they can zoom in on the desired value for x and y.

11 Summer 2006 I2T2 Probability & Statistics Page 147 See Writing Activities, questions 11 and 12 on Student Question sheet. One possible answer to #11 is that a long rectangular pen would give Fido more length to run. See Related Extension Activities, questions 13, 14, 15 on Student Question sheet.

12 Summer 2006 I2T2 Probability & Statistics Page 148 Student Questions for Fido's Fence Fido, our mutt, keeps running away and we need to fence him in. The fencing costs $3 a foot and we have only $126 to spend. How much fencing material can we buy? Fido is a very hyperactive dog. We need to build the largest possible rectangular pen so he has plenty of room to run. 1. How much fencing can we buy? How can we calculate this? 2. What do the dimensions of the pen need to be? a. Sketch the possible rectangles on the grid paper. b. In the cooperative group, compile all of the data on a chart. c. As a class compile all of the data into one chart. i. Look for patterns, compare width and length, compare dimensions with the area. ii. Discuss the maximum area of the dog pen. Are there solutions that are not whole numbers that would give us a bigger area? 2. How can you write an expression for the length, width, perimeter, and area in terms of the length? a. length = x b. width = c. perimeter = d. area = If L1 = length, then what would you enter for the formulas for the following? a. L2 = width = b. L3 = perimeter = c. L4 = area = 3. Using the lists, what is the maximum area of Fido's pen? What are the dimensions of the pen? 4. To see a graphical solution for this problem, which two lists should you use? Explain why. 5. What should the window settings be? Explain why. 6. Predict what the graph will look like based on the information in the lists.

13 Summer 2006 I2T2 Probability & Statistics Page Using TRACE compare the ordered pairs to the corresponding values in the lists. a. Discuss how well they predicted the shape of the graph. b. Why are there two points with the same area? c. Estimate the maximum area. What are the length and width for this area? d. Estimate the dimensions of the figure if the area is 75. (Use the arrow keys.) 8. If the length = x, then how can you represent the area as a function of x? Enter the area function in Y1. Graph this function along with the scatter plot. a. Press the TRACE key. Press or to change to the graph instead of the plot. b. What do the x and y represent in the display? c. Verify the maximum area from your previous estimation in 7c. d. Verify the dimensions of a figure with an area of 75 (from 7d). e. What happens to the area as the length approaches zero? f. What does the graph of the continuous function show us that the scatter plot cannot? 9. Use the table feature to view the relationship between the length and area. Set up the table and estimate the dimensions when the area is 75. a. Using the TBLSET allows us to get closer to the area approximation by changing the size of ΔTbl. Start with TblStart=1 and ΔTbl=1 and scroll to about 75. Keep changing these values until you get closer to Y1=75. b. Use [TABLE] to visualize the maximum area. 10. What will be the dimensions of Fido's dog pen? 11. What are some reasons for not building a pen that has a maximum area? 12. Your friend has asked you how to find the maximum area of a rectangle with the perimeter of 50. Write the directions for finding this.

14 Summer 2006 I2T2 Probability & Statistics Page After all of your work, you now found out that you can use the side of the house for one of the sides of the pen. What is the maximum area you can fence? What are the dimensions? 14. You have found fencing materials at You-Save-A-Lot Building Supply for $2.25 a foot. What is the maximum area of a rectangular pen you could build while still spending the same amount of money ($126)? What are the dimensions? 15. Fido still needs a pen and we have only 42 feet of fence. But now consider that the fence can bend. Does the rectangular pen still provide the maximum area given the fixed perimeter? (Should we consider a different shape?)

15 Summer 2006 I2T2 Probability & Statistics Page 151 Box It Up: Volume Adapted from Graphing Calculator Activities for Enriching Middle School Mathematics, Browning & Channell, Texas Instruments, Concepts/Skills: Maximum volume of an open box, dependent and independent variables. Needs: Graphing calculators, sheet of graph paper for each student or group of students, material to fill boxes to measure volume (puffed rice, popcorn, etc.), scissors, tape, rulers Getting started: Tell students the following story: Ms. Hawkins, the physical sciences teacher at Buffalo High School, needs several opentopped boxes for storing laboratory materials. She has given the industrial technologies class several pieces of metal sheeting to make the boxes. Each of the metal pieces is a rectangle measuring 30 cm. by 50 cm. The class plans to make the boxes by cutting equal-sized squares from each corner of a metal sheet, bending up the sides, and welding the edges. Procedure: To better understand this problem, have the class start by making paper models of the kind of box Ms. Hawkins has requested. a. Using graph paper, where each square represents 1 cm 2, mark off a rectangle 30 squares x 50 squares. b. Mark off equal-sized squares at each of the corners and then cut out the squares. c. Fold the side tabs up and tape each edge together to form an open-topped box. d. Determine whose box has the greatest volume. Fill each box with puffed rice, popcorn, or some other material. Pour into a large container and mark the volume of each box. Which box has the greatest volume? e. Compare the general size and shape of your box with the boxes made by others in your classroom. Note that the size of the square cut from each corner of the paper determines the dimensions of the final product. What happens to the height, length, and width of the box as the size of the cut squares gets larger? You can see that the industrial technologies class could make several differently shaped boxes. Students should answer questions 1-5 on the Student worksheet. See Student notes for directions for setting up the table. Answers to worksheet questions follow below in italics. 1. Height:5 cm, Length: 40 cm. Width: 20 cm, Volume: 4000 cm 3 2. Height: 2 cm, Length: 46 cm, Width: 26 cm, Volume: 2392 cm 3 3. Does cutting out and throwing away less sheet metal necessarily result in a box with a larger volume? Explain. from previous examples 4. What is the largest possible value for the length of the square? Explain why no larger value could be used. 14 cm

16 Summer 2006 I2T2 Probability & Statistics Page 152 Height (cm) H Length (cm) L Width (cm) W Volume (cm 3 ) V h 50 2h 30 2h (50 2h)(30 2h)h

17 Summer 2006 I2T2 Probability & Statistics Page Write a sentence that would describe the meaning of the two numbers in the first row of the table. When a square of 1 cm is cut out from each corner of the rectangular piece of metal, a box with a height of 1 cm and a volume of 1344 cm 3 can be produced. 8. Describe any patterns you see in the sequence of volumes as you continue to scroll to the value x = 14. For example, as the size of the cut squares increases, what happens to the volume of the box? The volume increases to a maximum value of 4104 cm 3 at x=6 cm. After that the volume decreases to a low of 616 cm 3 at x = 14 cm. If you continue to scroll past 14 cm, you get zero and negative volumes. 9. What size of square cut (value of x) gives a box with the largest volume (value of Y1)? Height: 6 cm, Length: 38 cm, Width: 18 cm, Volume: 4104 cm What are the dimensions and volume of the box that the industrial technologies class should make for Ms. Hawkins? Height: 6 cm, Length: 38 cm, Width: 18 cm, Volume: 4104 cm You should have found that the box, of largest volume, is made by cutting squares with sides of length 7 cm. However, this result assumes the height x is restricted to whole number values. What if we could now make cuts to tenths of a centimeter in the sheet metal? Between what two whole number heights do you believe the maximum volume would be found? (Do not have to be sequential numbers.) Write down those two values along with their corresponding volumes. X = 5, Y = 4000 and X = 7, Y = Discuss these choices in your group and provide an argument to show why the maximum volume must occur for some x between your two chosen values. The volume has been gradually increasing from X=1 to 6, then decreasing from X=6 to 14. So the maximum must be between 5 and Describe any surprises as you scrolled through the table of values. Students often anticipate that is will be halfway between 5 and 6 or 6 and What is the new best value for x? 6.1cm 15. What is the volume of the best box? cm Is the volume of this box significantly larger than the one with a height of 6 cm? no 17. In summary for this numerical look at volume, write three key points or ideas your group has found while determining the maximum volume for the boxes Ms. Hawkins needs for her physical sciences class. Sample ideas: Volume of box increases and decreases as you increase the height; we were able to find the volume knowing only one dimension; we were able to write an equation relating height and volume; relationship between height and volume is not linear.

18 Summer 2006 I2T2 Probability & Statistics Page 154 Extension A graphical look at Box It Up entails using the same equation, setting the window, then analyzing the graph, zooming in on the maximum point of the graph. See student notes for directions. Answers are in italics below. 1. Why must x have a value between 0 and 15? Because 1 cm is the smallest cut and at 15 the box has no volume. 4. What are the values of x and y at the highest point on the graph? x= , y= Record these x-coordinates , Record the coordinates from the screen. Sample values: x= , y= By observing the x-coordinates of the points on either side of this point, you can again determine an interval that contains the desired solution. What x-interval contains the value associated with the maximum height of the graph? Sample interval (to nearest hundredth) is 6.04, Write the new interval for the maximum volume. Determine whether you can provide an x-value accurate to the nearest tenth. Remember, you want all numbers in your interval to round to the same tenth s value. Sample interval is 6.064, Based on the results from your trace of the graph, what size squares should the class cut from the pieces of sheet metal? What are the dimensions and volume of the box with the largest volume? Side of square:6.1. Height: 6.1. Length: Width: 17.8, Volume: You have now investigated the box volume problem using two methods: tabular and graphical. Discuss in your groups the advantages of both methods. Write those advantages below. Some prefer the tabular approach with the ΔTbl feature making zooming in fairly easy. Others prefer a picture and would rather zoom in on the graph for a maximum value. Sometimes when viewing a table of values, it is difficult to get a sense of how the function behaves over a larger interval; the graph provides a bigger picture. One can find a maximum value fairly quickly on the table without having much sense about how the function behaves. When creating a graph you have the additional challenge of defining the window, which requires some knowledge of the behavior of the function.

19 Summer 2006 I2T2 Probability & Statistics Page 155 Student Worksheet for Box It Up Ms. Hawkins, the physical sciences teacher at Buffalo High School, needs several opentopped boxes for storing laboratory materials. She has given the industrial technologies class several pieces of metal sheeting to make the boxes. Each of the metal pieces is a rectangle measuring 30 cm by 50 cm. The class plans to make the boxes by cutting equal-sized squares from each corner of a metal sheet, bending up the sides, and welding the edges. To better understand this problem, start by making paper models of the kind of box Ms. Hawkins has requested. a. Using graph paper, where each square represents 1 cm 2, mark off a rectangle 30 squares x 50 squares. b. Mark off equal-sized squares at each of the corners and then cut out the squares. c. Fold the side tabs up and tape each edge together to form an opentopped box. d. Determine whose box has the greatest volume. Fill each box with puffed rice, popcorn, or some other material. Pour into a large container and mark the volume of your box. Which box has the greatest volume? e. Compare the general size and shape of your box with the boxes made by others in your classroom. Note that the size of the square cut from each corner of the paper determines the dimensions of the final product. What happens to the height, length, and width of the box as the size of the cut squares gets larger? You can see that the industrial technologies class could make several differently shaped boxes.

20 Summer 2006 I2T2 Probability & Statistics Page 156 Questions 1. If the class decided to cut 10-cm squares from the corners of the metal sheets, determine the dimensions and volume of the box that could be formed from the remaining sheet metal. Size of square: 5 cm Height: cm Length: cm Width: cm Volume: cm 3 2. What would be the dimensions and volume of the box if 5-cm squares were cut from the corners of the metal sheets? Size of square: 2 cm Height: cm Length: cm Width: cm Volume: cm 3 3. Does cutting out and throwing away less sheet metal necessarily result in a box with a larger volume? Explain. 4. Squares of many different sizes can be removed from the corners of the metal sheets. If cuts were made in whole centimeter lengths, the smallest square would have a side of length 1 cm. What is the largest possible value for the length of the square? Explain why no larger value could be used. 5. Ms. Hawkins decides she wants to have boxes made with the largest possible volume. How could the class determine which size square to cut out so that a box with the largest volume is produced? If you enter the information you found earlier in a table, it may help you find a relationship between the dimensions of the box. Notice that the size of the cut-out square determines the height of the box. Complete all but the last line of the table.

21 Summer 2006 I2T2 Probability & Statistics Page 157 Height (cm) H Length (cm) L Width (cm) W h Volume (cm 3 ) V Since you can choose the height of the box, H, height is called the independent variable. The other dimensions depend upon your choice of H, so they are dependent variables. When you pick H, the resulting length, L, is 50 2H, for example 50 cm 2(5 cm) = 40 cm. The width, W, is found by using the expression 30 2H. Finally, the volume, V, is the product of all three dimensions, of L*W*H. Substitute in the other expressions for L and W in the product for finding volume if the height = h: V = L*W*H = ( )( )h. Complete the last row of the table using the expressions you have found for L, W, and V in terms of h. 6. How would you calculate the volume of a storage box if you knew only the height of the box?

22 Summer 2006 I2T2 Probability & Statistics Page 158 Using the Calculator You can use the graphing calculator to generate a table of values. Then the box with the largest volume can be found by searching through these values. In order to generate a table of values, you must provide the calculator with three pieces of information: An expression written using only one variable, x. The value of x to begin the table calculations. The size of the step to be used in moving from one value of x to the next. For this situation, the expression we have written is (50 2h)(30 2h)h. We need only to refer to the variable height as x instead of h. So, V = (50 2x)(30 2x)x. We decided that the smallest cut-out square would have a side length of 1 cm, which means that the table should start at x = 1. The value of x should increase by 1 each time the next table entry is calculated, so the table will contain whole number values for the cuts. a. Press Y=. If there are any expressions listed, press the arrows to each line and press CLEAR. If any Plots are highlighted, arrow up to them and press ENTER to turn them off. Enter the expression (50 2x)(30 2x)x for the volume of the box. b. Press 2 nd [TblSet] to access the Table Setup window. Enter the TblStart = 1, the smallest cut. Enter the ΔTbl = 1, the increase in the x value. Make sure that Auto is highlighted on the last two lines. c. Press 2 nd [TABLE] to see the first few rows of the table. 7. Write a sentence that would describe the meaning of the two numbers in the first row of the table.

23 Summer 2006 I2T2 Probability & Statistics Page 159 Even though the window display shows only 7 lines of the table, the up-down arrows can be used to move up and down the display. Use the down arrow to move past the last line in the display. 8. Describe any patterns you see in the sequence of volumes as you continue to scroll to the value x = 14. For example, as the size of the cut squares increases, what happens to the volume of the box? 9. What size of square cut (value of x) gives a box with the largest volume (value of Y1)? Height: cm Length: cm Width: cm Volume: cm What are the dimensions and volume of the box that the industrial technologies class should make for Ms. Hawkins? Height: cm Volume: cm 3 Length: cmwidth: cm 11. You should have found that the box, of largest volume, is made by cutting squares with sides of length 7 cm. However, this result assumes the height x is restricted to whole number values. What if we could now make cuts to tenths of a centimeter in the sheet metal? Between what two whole number heights do you believe the maximum volume would be found? (Do not have to be sequential numbers.) Write down those two values along with their corresponding volumes. X = X = Y = Y = 12. Discuss these choices in your group and provide an argument to show why the maximum volume must occur for some x between your two chosen values. Set TblStart to the smaller of the two X numbers you argued for above. Set ΔTbl to.1 so that the change between x values will not be tenths of a centimeter. Construct a new table. 13. Describe any surprises as you scrolled through the table of values. 14. What is the new best value for x?

24 Summer 2006 I2T2 Probability & Statistics Page What is the volume of the best box? 16. Is the volume of this box significantly larger than the one with a height of 6 cm? 17. In summary for this numerical look at volume, write three key points or ideas your group has found while determining the maximum volume for the boxes Ms. Hawkins needs for her physical sciences class. a. b. c. A Graphical Look at Box It Up You have already built a table showing the volumes that resulted when squares of certain sizes were removed. Another approach often used to solve such problems is to generate a graph of the relationship between box height and volume on a coordinate grid. To graph a relationship using a graphing calculator, it is necessary to carry out two tasks: Define the relationship in terms of two variables, x and y. Define the portion of the coordinate plane over which you wish to view the graph. If you have not cleared your Y= screen, you should still have Y1 defined as (50 2x)(30 2x)x. That is the first task. The second task is to define the portion of the coordinate plane that you wish to view, referred to as the window on the calculator. Press the WINDOW key. The values for Xmin and Xmax are the left and right endpoints of the viewing window. The values for Ymin and Ymax are the lower and upper limits of the viewing window. Xscl and Yscl are used to define the distance between the tick marks on the two axes; they do not affect the viewing window or appearance of the graph.

25 Summer 2006 I2T2 Probability & Statistics Page 161 Now define the limiting values for the viewing window considering the equation (50 2x)(30 2x)x and think about the values used to answer the problem using the table. Given the conditions of the problem, x must take on the values between 0 and Why must x have a value between 0 and 15? If tick marks are desired, you can set Xscl to 1 and have tick marks at every unit along the x-axis. Defining the limits on y is a little more challenging. A given value of Y1 represents the volume of a box for some height x. Volume cannot be negative, so you can define Ymin to be slightly less than 0. You are looking for the x that would produce the largest value of y possible, but this doesn t indicate how large y can get. However, your previous work with tables showed that all volumes were less than 4200 cubic centimeters, so a reasonable value for Ymax is 5000 (to give room to display the function at the top of the screen). Since the height of this view is very large, the Yscl should be large. Approximately 20 tick marks per axis is about right, and 5000/20 = 250, so enter 250 for Yscl so that tick marks on the y-axis are placed at 250, 500, 750,, When entering negative values, be certain to use the (-) key and not the minus key. Enter the following values into WINDOW. Xmin = 1, Xmax = 17, Xscl = 1, Ymin = 10, Ymax = 5000, Yscl = 250 Press the GRAPH key. A graph similar to the one on the right should appear. 2. In your group, write a brief explanation of why the graph should look the way it does based on your previous work with the volume table and the patterns you observed there.

26 Summer 2006 I2T2 Probability & Statistics Page 162 Press the TRACE key once to read coordinates of points on this graph. Note that the blinking crosshair and the coordinates are displayed at the bottom of the screen. This informs you that the point where x = 8 and y = 3808 lies on this view of the graph. 3. Do the coordinates of points on the graph agree with the values you computed earlier in this activity? Explain what these two values mean with respect to the box volume problem. Use the left and right arrows to move the blinking crosshair. The crosshair will move only along the graph, as long as you are in TRACE, and not just anywhere in the coordinate plane. Notice that the coordinates at the bottom of the screen change as you press the arrow keys. The x-coordinates are determined by the calculator based upon the current values of Xmin and Xmax. The y-coordinates are computed using the expression (50 2x)(30 2x)x. You will probably see that the x-values are not always nice numbers, unlike those you saw when you generated the tables. Move the cursor until you locate the point on the graph that gives the largest y-coordinate. 4. What are the values of x and y at the highest point on the graph? x = y = Not all of the decimal places shown in the coordinates of this point are meaningful. In fact, this point may not be the one you are looking for. The screen is restricted by the number of pixels that can be shown, so this point is the closest point on this view of the graph. By examining the x-coordinates of the points on each side of this point, you can find an interval that contains the best value of x. Use the arrow keys to determine the x-coordinates of the points immediately to the left and to the right of the point found above. 5. Record these x-coordinates. Point to the left: Point to the right: The solution you are looking for lies somewhere between these two x- values. Check the table of values to see that the tabular solutions fall somewhere between the two values above. (Press 2 nd [TABLE] to access the table of numbers.)

27 Summer 2006 I2T2 Probability & Statistics Page 163 Just as we did when we used the tables to investigate this problem, it is easy to examine the volume function over a smaller interval in order to obtain a more precise approximation to the desired solution. This can be done in a number of ways, but the fastest is to use ZOOM. Press the ZOOM key and consider the two options Zoom In and Zoom Out. Zooming in is similar to bringing a portion of your current view closer for a finer examination. Zooming out is equivalent to enlarging your field of view so that you see more of the graph, but less detail. Since you want a closer, more detailed look at the points near the apparent maximum if the graph, you must zoom to the region containing that maximum. A quick way to do this is to use the option ZBox. ZBox lets you use the cursor to select opposite corners of a box to define the portion of the current view that you wish to enlarge. Press ZBox. You will see a different crosshair in the center of the screen. Use the arrow keys to move the cursor somewhere above and to the left of the highest point of the graph. Once you are satisfied with the cursor s location, press ENTER. Now move the crosshair to the right and down until you reach the opposite corner of the box you are forming. As you do this, you will see the box forming on the screen. Try to center the point of interest in the box. One possible zoom box is shown on the right. When you are satisfied with the location of the second corner, press ENTER again. The calculator should redraw the graph in the window you have just defined. Use TRACE and the arrow keys again to locate where you believe the maximum value to be. 6. Record the coordinates from the screen. x = y = 7. By observing the x-coordinates of the points on either side of this point, you can again determine an interval that contains the desired solution. What x-interval contains the value associated with the maximum height of the graph? If the tenths position in the x-interval is not the same for both values, or if both x-values when rounded to the nearest tenth are not the same, then the best choice for x we can make has to be a whole number. If you zoom one more time on the place where you believe the maximum volume will be, you will be able to provide a value for x that is accurate to the nearest tenth.

28 Summer 2006 I2T2 Probability & Statistics Page 164 Press ZOOM and select ZBox again. Make another box around the area on the graph where you think the maximum volume lies. Once you ve made the box, press ENTER. A new graph appears looking something like the graph displayed. (Remember, we haven t necessarily chosen identical zoom boxes.) Press TRACE to find new x-values for an interval containing the maximum volume as you have done previously. 8. Write the new interval for the maximum volume. Determine whether you can provide an x-value accurate to the nearest tenth. Remember, you want all numbers in your interval to round to the same tenth s value. 9. Based on the results from your trace of the graph, what size squares should the class cut from the pieces of sheet metal? What are the dimensions and volume of the box with the largest volume? Side of square: Length: Volume: Height: Width: 10. You have now investigated the box volume problem using two methods: tabular and graphical. Discuss in your groups the advantages of both methods. Write those advantages below. 11. Which method do you prefer and why?