The Apple Orchard Problem - uakron.edu€¦ · Web viewThe Apple Orchard Problem. Lesson Summary:...

The Apple Orchard Problem

Lesson Summary: In this activity, students will explore various approaches to solving the problem of maximizing

the total production of apples in an orchard using several techniques on the TI-Nspire CAS calculator.

Key Words:Quadratic Regression, First Derivative Test, Second Derivative Test, Critical Points

Background Knowledge:This activity will have the students exploring the maximum number of trees to add to an orchard

before production starts to decline. Students should have a general understanding of how to find the minimum and maximum of functions using a variety of methods. Some of these include interpretation from a table, using a graphical approach, and using calculus. Students should be somewhat familiar with the key words mentioned above, and understand how to apply them. They also must have a basic knowledge of how to use the TI-Nspire CAS handheld components such as Calculator, Lists & Spreadsheet, and Graphs & Geometry.

Materials: TI-Nspire CAS handheldWorksheetComputer

Suggested Procedure: Students will be put into groups of two or three. The students will then open the file on their

computer that will guide them through the activities. There are questions that will provide a place to record their observations. Remind students that they are trying to maximize the production of apples at the orchard. Discuss and review how to use different applications on the TI-Nspire CAS handheld, such as Calculator, Graphs & Geometry, Lists & Spreadsheet, and Notes. Instruct the groups to open the activity document, and have the teams complete the activity.

Standards: Patterns, Functions and AlgebraB. Students use patterns, relations and functions to model, represent and analyze problem situations

that involve variable quantities. D. Students analyze, model and solve problems using various representations such as tables, graphs

and equations.

Benchmark/Grade Level Indicator: Use Patterns, Relations and Functions4. Identify the maximum and minimum points of polynomial, rational and trigonometric functions

graphically and with technology.

Assessment: Check students progress in class on the TI-Nspire CAS handheld. Have the students submit their

completed activities.

***Note: The TI-Nspire CAS calculator at this time is a new piece of technology for many students. In the future students will be more knowledgeable with the TI-Nspire. Throughout the lesson, comment boxes explaining how to complete procedures are included. In the future, once students are comfortable with the TI-Nspire, the comment boxes may be removed.

Project AMP A Quesada Director Project AMP

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Activity 1: Data Collection and Analysis. In this activity, you will be required to collect data in order to solve the Apple Orchard problem.

Goal: Students will discover the number of trees that should be added to maximize the apple production. They will do this using a table.

Students’ Names: _____________________________________

Apple Orchard Problem: There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees?

1. Start by creating a new document. This can be done by pressing Home 5:New Document or ctrl N.

2. We want to create a spreadsheet.When prompted by the calculator, choose 3:Add Lists and Spreadsheets.

3. Next, we want to create a spreadsheet that will help us in determining how many apple trees we should plant in order to maximize the number of apples. In order to do this, we must have three different columns in our spreadsheet. In column A, we want to type “trees” to represent the total number of trees. This can be done by simply pressing the up button twice, until it is directly beside the letter A. When it is here, type in the phrase “trees.” After you complete this, hit the enter button.

4. Now we want to resize the column so we can see our heading. To do this, hit the Menu button. Then go to 1:Actions, 2:Resize. This will highlight column A. Now, use the right arrow button to move the column width until you can view all of your heading. Then hit enter.

5. Next, we need to create two more columns. Complete the previous steps to add the phrase “apples” (to represent the total number of apples per tree) in column B and “total” (to represent the total number of apples produced) in column C. The resulting sheet should look like the following:

6. We know the starting conditions for this optimization problem. They are: 50 trees that produce 800 apples each. Thus, we can enter this into our data. Go to cell A1 (use the arrow keys until it is outlined). Hit enter. Now, enter in the number of trees, which is


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50, and hit enter. Next, we want to move to cell B1 and enter the number of apples, which is 800.

7. Next, we want to determine the total number of apples produced. Create a function that will perform the calculation for you.We can define a function to perform the calculations. Use the arrow buttons to move to the cell that is below the box C and total. Hit enter on this box. It will show total:= What we want to do is define the function we are going to use. In this box, type “trees”, then hit the multiplication button, then type “apples”, and hit enter. Take a second to resize your column C to allow you to see the entire function created. This can be done by the following keystrokes: Menu, 1:Actions, 2:Resize. This is what your viewing window should look like:

8. Take the next few minutes filling in different values in the table. (The calculator may give you an error when you input the first value for trees. This is because there is a dimension error for your function. Hit enter whenever this happens and continue entering the data.) What happens after you input the number of trees and apples into your spreadsheet?

When the data is being filled in, the corresponding cell in the spreadsheet is automatically updated. This is because the math formula automatically updates when values are entered.

9. Based on the data that you have collected, how many trees should be added in order to maximize apple production (be careful with interpretation)? What is the maximum number of apples that you found?

The number of trees that should be added is 15. The total number of apples produced will be 42,250 apples.

Extension:


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1. Can you think of any other situation where collecting data would be helpful in solving a complex optimization problem? List the problem and explain how you would collect the data. (You may want to conduct research on the Internet)

There can be a wide range of answers here. I will provide two as a basis.

Example One: Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum. In this problem, the students can create a table with columns consisting of relevant information. Then, the students will collect the data using different numbers.Example Two: Find the length of the shortest ladder that will reach over an 8-ft high fence to a large wall which is 3 ft behind the fence. (see diagram) What the students can do here is construct this figure on a sheet of paper, using the scale inches instead of feet. Then, the students can create different ladders and measure the lengths with a rule. They can record the data in a table and determine the shortest ladder.


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Activity 2: Determining a line of regression.

Goal: The students will plot the data and look at the graph. They are going to do this to determine the line of regression for the data.

1. Start by inserting a new page so we can graph the data. Do this by pressing ctrl I. Then, choose 2:Add Graphs & Geometry.

2. We want to make a scatter plot of the data that we have collected, so we need to change the graph type. Press menu. Then, press 3: Graph Type, 3: Scatter Plot.

3. Next, we need to determine the independent variable and the dependent variable. We want to plot the number of trees added in relation to the total number of apples produced. What is the independent variable? The dependent variable?

The independent variable is the number of trees that we added to the starting amount of 50. The dependent variable is the total number of apples produced.

We need to make a new column that represents the number of trees added. Why must we plot this value instead of the total number of trees?

If we use the values 50, 51, etc, it does not represent the number of trees being planted. Instead, it represents the total number of trees. This is not what we are looking for.

4. Go back to the spreadsheet and insert a new column between trees and apples, and label it “added.” This column will represent the number of trees added from the initial starting point. Start by pushing the ctrl, left arrow button switch pages. Next, we can insert the new column. This can be done by moving the cursor onto the label B of the column. Then, hit Menu, Insert, Insert Column. Now fill in the corresponding values.

5. Now, let’s graph the scatter plot. Make sure the box beside the x is highlighted, and hit enter. Then, choose the list “added” by using the arrow buttons and hitting enter.

6. Press the tab button until the box next to the y is highlighted (it should only take one press). Hit enter and find the list “total”.

7. Now, we must configure our viewing window to see the scatter plot. Look back at your data. You can do this by hitting ctrl and the left arrow button. Determine the following values that correspond to your data:

There is a range of acceptable values that can be used. I will just give an example.x min: -1 x max: 30y min: 37500 y max: 42500

8. Next, hit ctrl and the right arrow button to get back to the graph.9. Hit the menu button and pick 4: Window. Then choose 1: Window Settings. Then,

type in the value for the x-min. Be sure to press the down arrow button when this is


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entered. Do NOT hit the enter button. Continue filling in the values. When completed, hit enter.

10. You should see the scatter plot of the data. It should look like the following:

11. Looking at the data, what do you think is the best possible model for this graph?

I believe that the best model for this graph would be the quadratic model. The reason for this is it appears to be a parabola.

12. Based on the model that you think would be the best fit, determine the parent function.

Parent Function: x 2

13. Next, we want to fit a graph to the data. Press menu. Then, press 3: Graph Type, 1: Function.

14. We want to insert your parent function into the f1(x). However, if we put it in now, it will not show up on our window. Therefore, we must conduct some translations to it. What do you think would be acceptable translations/reflections to make the graph show up on your new window and look like the data?

There would be two things that we should do to the parent function. We should shift the function up 42,000. We also should reflect it about the x-axis.

15. Let’s use the quadratic as our model. Thus, our parent function would be y = x².16. Next, enter the translated equation into f1(x).17. Thus, we get the following:


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18. Push the tab button until there is a pointer in the graphing window. Use the arrow buttons to move the cursor onto the function f1(x) until a icon appears that looks like the following in the upper right corner:

19. Click and hold the center button located on the wheel until it looks like a closed fist. Now, use the arrow buttons to make it look like the shape of the data. When you have a rough estimate, hit the center button again.

20. Next, move the cursor to the vertex of the quadratic. It will change into a crosshair. When this happens, click and hold the center button on the wheel. Then, move the vertex so it matches with the data.

21. Continue this process until you get a graph that is representative of the data. It should be roughly similar to the following:


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22. Now, look at the function that you have for f1(x). Make an educated guess at what you believe the equation for this model will be.Estimation: -10x 2 + 300x + 40000 (any guess within reason would be acceptable)

23. Let’s check if we got the correct regression model. Perform the quadratic regression on your data. What did you get?

-10x 2 + 300x + 40000

How to perform quadratic regression. Start by creating a new page. Hit ctrl I, 1: Add Calculator. Then, hit Menu, Statistics, Stat Calculations…, Quadratic Regression. When prompted, insert “added” into the X List and “total” into the Y List. Hit enter.

Extension:1. Can you think of any different situations where quadratic regression and other types of

regression are used for predictions?

There can be a wide range of answers for this extension question. Here are a few examples.

Exponential Regression is used in most all growth models.Quadratic Regression is used when there is a projectile.


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Activity 3: Algebraic Approach

Goal: The students will use the TI-Nspire CAS to graph the function that corresponds to the data and determine the local extrema (number of trees to maximize production).

1. Let the variable x be the additional number of trees that we are going to plant in the orchard. We are trying to maximize the total production of apples. Let P represent the total production. Then,

P = (50 + x )(800 – 10x) = -10x 2 + 300x + 40000 (Be sure to write P in factored form and expanded form.)

2. Next, we want to graph this function. Insert what you discovered for P in f1(x).3. Now, we must configure our viewing window to see the graph of the function. Start by

plugging numbers into the equation to determine an acceptable viewing window. Determine the following values of the window that corresponds to your data:

There is a range of acceptable values that can be used. I will just give an example.x min: -1 x max: 30y min: 37500 y max: 42500

4. Next, determine the maximum. Give the values. In your own words, interpret your findings.

The maximum number of trees that should be added is 15 trees. This will result in a total production of 42,250 apples. This means that if you plant any more trees, or any fewer trees, then the apple production will be below 42,250 apples.

To find the maximum, push Menu, Points & Lines, Point On. Then, click anywhere on the graph of the function. Next, hit esc. Then, move the cursor onto the point and click and hold the center button. Finally, move the point until an M shows. This represents the maximum value.


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Extension:

1. How can we check to make sure our solution is the number of trees that maximizes the apple production?

We can plug our point back into the function and make sure it is a maximum by then picking points to the left and right of our solution. This will show if the production of apples is lower than at the solution.

2. Based on your own knowledge, is there a more advanced way to solve this problem? If so, explain what you would do.

We could solve this using calculus. We could conduct the first and second derivative tests. This is tell us many valuable things as to what the number of trees added will result in the maximum number of apples produced.


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Activity 4: Dynamic Geometry and Data Capture

Goal: The students will use the TI-Nspire CAS to create a graphical representation of the problem. They will then automatically collect data, obtain a scatter plot, and determine the best fitting regression line. Students will then be able to determine the number of apple trees that should be added in order to maximize production.

1. Start by inserting a new page so we can create a graphical representation for the Apple Orchard problem. Do this by pressing ctrl I. Then, choose 2:Add Graphs & Geometry.

2. In order to create the graphical representation with the dynamic geometry, we must determine a function. This will be a linear function consisting of the original conditions for the problem. Complete the following:

Number of initial apple trees: 50 treesNumber of apples each tree produces: 800 applesFor each tree added, output per tree drops: decreases 10 apples per tree

3. Using these values, determine a linear equation. (Note: Be careful with the slope)

Y = -10x + 1300

4. Based on your linear equation, determine when the equation will equal zero.

x = 130

5. Determine the viewing window that we will use for our graph.

There is a wide range of acceptable answers, here is one example:x min: -20 x max: 150y min: -50 y max: 1500

6. Now, graph your function with the viewing window. It should look like the following:


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7. Next, we want to create a point on the x-axis with the coordinate (50,0). Label the point “A”.This can be done with the following keystrokes: Hit menu, 6:Points & Lines, 2: Point On. Now move the cursor to the location on the x-axis where 50 is located. It is easiest to do this if your scale for the x-values is 10. Hit enter. If the coordinates are not being displayed, you can hit menu, 1: Tools, 6: Coordinates and Equations, and then click the point. This can also be used for equations of lines. You may need to do this in the future, so use this as a reference. To label a point, hit menu, 1: Tools, 5:Text, and then click on the point you wish to label.

8. Next, create a line that contains the point and is perpendicular to the x-axis.Press menu, 9: Construction, 1: Perpendicular. Then, click on the point and the x-axis

9. What is the equation of this line? x = 50

10. Find the point of intersection of the perpendicular and your equation and label this point “B”.To do this, press menu, 6: Points & Lines, 3: Intersection Point(s). Move the curser to the point of intersection. A point will appear. Hit enter. If you cannot get the point to appear, you can also just pick the two lines that intersect.

What are the coordinates of the point? (50, 80)

11. Now, create a line through the newly create point that is perpendicular to the y-axis.Press menu, 9: Construction, 1: Perpendicular. Then, click on the point and the y-axis

12. What is the equation of this line? y = 800

13. What is formed in relation to the x-axis, y-axis, and the lines found in steps 9 and 12?

The figure that is formed is a rectangle.

14. Determine the area of this region. What does it represent?

The area is 40,000. This represents the total number of apples produced.

Here is what your viewing window should look like:


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15. Next, we want to collect data to determine a regression equation. Before we can do this, we must first create another point. Create a point at the intersection of the line y = 800 and the y-axis. Label this point “C”.

16. Determine the length of AB and BC. What does AB and BC represent?

The length of AB represents the number of apples that are produced for each tree. The length of BC represents the total number of trees being planted.

Hit menu, 7: Measurement, 1: Length. Click the Points A and B. Then hit enter.17. Store these values into the variables “AB” and “BC” respectively.

Click on the measurement. Then press ctrl, var. Next, type in the name of the variable, and then hit enter.

18. Create a list for the values collected. Choose ctrl I to insert a new page. Choose 3: Lists and Spreadsheets. To name the lists, place the Pointer in the white space to the right of the shaded title of the column and enter a name, “Length”. Repeat this for the other column to match the number of variable values you wish to collect, “Width”. The list names must be different then variable names.

19. Automatically capture the data into your list.Move the cursor into the formula cell just below the column name. Press menu, 3: Data, 2: Data Capture, 1: Automated Data Capture, and then press var, 3: Link To, and select the variable from the list and press enter. For Length, use BC, and for Width use AB.


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20. Collect several other data points.Press ctrl and the left arrow to return to the page with our graph. Grab point A and drag it to the right. Be sure to go slow so you do not skip any data points. It will collect the points on an interval of 10. It is doing this because this is our scale.

21. Return to the spreadsheet and look at the data points.22. Now, we want to create a column that represents the area. What does the area represent

in this problem?

The area represents the total number of apples being produced.

Refer to step 18 to create a column named “Area”.23. In the function box for the column Area, add the function length times width.

24. Next, we want to make a scatter plot of the data we have. Insert a new Graphs and Geometry page. Change the graph type to scatter plot.Press menu. Then, press 3: Graph Type, 3: Scatter Plot.

25. Next, we need to determine the independent variable and the dependent variable. We want to plot the number of trees added in relation to the total number of apples produced. What is the independent variable? The dependent variable?


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The independent variable is the number of apple trees being planted, and the dependent variable is the total number of apples being produced.

We need to make a new column that represents the number of trees added. Why must we plot this value instead of the total number of trees?

If we use the values 50, 60, etc, it does not represent the number of trees being planted. Instead, it represents the total number of trees. This is not what we are looking for.

26. Go back to the spreadsheet and insert a new column between length and width, and label it “added.” This column will represent the number of trees added from the initial starting point. Start by pushing the ctrl, left arrow button switch pages. Next, we can insert the new column. This can be done by moving the cursor onto the label B of the column. Then, hit Menu, Insert, Insert Column. Now fill in the corresponding values.

27. Now, let’s graph the scatter plot. Make sure the box beside the x is highlighted, and hit enter. Then, choose the list “added” by using the arrow buttons and hitting enter.

28. Press the tab button until the box next to the y is highlighted (it should only take one press). Hit enter and find the list “area”.

29. Now, we must configure our viewing window to see the scatter plot. Look back at your data. You can do this by hitting ctrl and the left arrow button. Determine the following values that correspond to your data:

There is a wide range of acceptable answers. Here is an example:x min: -10 x max: 90y min: -5000 y max: 46200

30. Next, hit ctrl and the right arrow button to get back to the graph.31. Hit the menu button and pick 4: Window. Then choose 1: Window Settings. Then,

type in the value for the x-min. Be sure to press the down arrow button when this is entered. Do NOT hit the enter button. Continue filling in the values. When completed, hit enter.

32. The scatter plot should look like the following:


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33. Create a new Calculator page and determine the best regression based on your data. Be sure to store the equation in one of the function possibilities.To perform regression, click menu, 6: Statistics, 1: Stat Calculations…

Type of Regression: Quadratic Regression

Equation for Regression: y = -10x 2 + 300x + 40000

(Note: your calculator may have some round off error. Round the coefficients to the nearest whole number.)

34. Using your line of regression, determine the maximum number of apple trees that should be planted in order to maximize production.

Number of apple trees to plant: The orchard owner should plant 15 apple trees.

Total number of apples produced: This will result in 42,250 apples.

To find the maximum, push Menu, Points & Lines, Point On. Then, click anywhere on the graph of the function. Next, hit esc. Then, move the cursor onto the point and click and hold the center button. Finally, move the point until an M shows. This represents the maximum value.


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Activity 5: Calculus Approach

Goal: the students will use the TI-Nspire CAS handheld to perform calculus operations to allow them to solve the Apple Orchard Problem.

1. Let the variable x be the additional number of trees that we are going to plant in the orchard. We are trying to maximize the total production of apples. Let P represent the total production. Then,

P = (50 + x )(800 – 10x) = -10x 2 + 300x + 40000 (Be sure to write P in factored form and expanded form.)

2. Differentiate the function P using the calculator.

P'= -20x + 300

This is the operator that you should use:

To differentiate, push menu, Calculus, Derivative. Next, fill in the variable and the expression and hit enter.

3. Next, we must determine the critical point(s) using the calculator. How do we determine what the critical point(s) are?

We determine the critical point(s) by setting the first derivative equal to zero and solving for the variable. The critical point(s) are also the points that make the derivative undefined. (Note, this will not happen in this problem so it is not necessary to receive all of the points for this question).

Critical point(s): x = 15

This is the operator that you should use:


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To get the critical points, push menu, Algebra, Solve. Then, enter the expression consisting of the function equal to zero. Then, push the “,” and put in the variable. Then, hit enter.

4. In your own words, interpret what P' and the critical point(s) tell us.

The derivative tells us the rate of growth of the apple production. The critical point tells us when the rate of apple production reaches zero. This indicates that the growth rate has reached a maximum, and is now starting to decline.

5. Create a sign chart using this information.

6. What can you conclude from the chart? Interpret your results.

From taking a quick glance at the sign chart, we know that there is either a maximum or a minimum at x=15. To the left of fifteen, the sign of the derivative is positive. To the right of fifteen, the sign of the derivative is negative. This indicated that x=15 is a maximum.

Extension:

1. Conduct the second derivative test.

P'' = -20


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2. Why is this helpful?

This is very helpful because it tells us the concavity at every point. Since the value of the second derivative is negative, it means that the function is concave down over its domain.

3. Interpret these results.

Since this function is concave down, and the critical point is x=15, we can conclude that this is a maximum of the equation.


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