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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 handheld
Worksheet
Computer
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 Algebra
B. 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 Functions
4. 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.
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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?
________________________________________________________________________
________________________________________________________________________
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?
________________________________________________________________________
________________________________________________________________________
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)
________________________________________________________________________
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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?
________________________________________________________________________
________________________________________________________________________
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?
________________________________________________________________________
________________________________________________________________________
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:
x min: ___________ x max: ___________
y min: ___________ y max: ___________
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
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:
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11. Looking at the data, what do you think is the best possible model for this graph?
________________________________________________________________________
________________________________________________________________________
12. Based on the model that you think would be the best fit, determine the parent function.
Parent Function: _____________________________________
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?
________________________________________________________________________
________________________________________________________________________
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:
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:
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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:
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: ____________________________________________________________
23. Let’s check if we got the correct regression model. Perform the quadratic regression on
your data. What did you get?
________________________________________________________________________
________________________________________________________________________
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.
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Extension:
1. Can you think of any different situations where quadratic regression and other types of
regression are used for predictions?
________________________________________________________________________
________________________________________________________________________
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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 = ________________________________________________
(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:
x min: ___________ x max: ___________
y min: ___________ y max: ___________
4. Next, determine the maximum. Give the values. In your own words, interpret your
findings.
________________________________________________________________________
________________________________________________________________________
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.
Extension:
1. How can we check to make sure our solution is the number of trees that maximizes the
apple production?
________________________________________________________________________
________________________________________________________________________
2. Based on your own knowledge, is there a more advanced way to solve this problem? If
so, explain what you would do.
________________________________________________________________________
________________________________________________________________________
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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: _________________________
Number of apples each tree produces: ______________________________
For each tree added, output per tree drops: __________________________
3. Using these values, determine a linear equation. (Note: Be careful with the slope)
__________________________________________________________________
4. Based on your linear equation, determine when the equation will equal zero.
__________________________________________________________________
5. Determine the viewing window that we will use for our graph.
x min: ___________ x max: ___________
y min: ___________ y max: ___________
6. Now, graph your function with the viewing window. It should look like the following:
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
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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? _____________________________________________
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? ________________________________________
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? _____________________________________________
13. What is formed in relation to the x-axis, y-axis, and the lines found in steps 9 and 12?
________________________________________________________________________
14. Determine the area of this region. What does it represent?
________________________________________________________________________
Here is what your viewing window should look like:
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?
________________________________________________________________________
________________________________________________________________________
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
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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.
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?
________________________________________________________________________
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.
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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?
________________________________________________________________________
________________________________________________________________________
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?
________________________________________________________________________
________________________________________________________________________
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:
x min: ___________ x max: ___________
y min: ___________ y max: ___________
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: ____________________________________________________
Equation for Regression: _________________________________________________
(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: ________________________________________
Total number of apples produced: ______________________________________
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 = ________________________________________________
(Be sure to write P in factored form and expanded form.)
2. Differentiate the function P using the calculator.
P'= ___________________________________
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?
________________________________________________________________________
________________________________________________________________________
Critical point(s): __________________________________________________________
This is the operator that you should use:
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.
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4. In your own words, interpret what P' and the critical point(s) tell us.
________________________________________________________________________
________________________________________________________________________
5. Create a sign chart using this information.
6. What can you conclude from the chart? Interpret your results.
________________________________________________________________________
________________________________________________________________________
Extension:
7. Conduct the second derivative test.
P'' = ________________________________________________________
8. Why is this helpful?
________________________________________________________________________
________________________________________________________________________
9. Interpret these results.
________________________________________________________________________
________________________________________________________________________
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