Inquiry_Lesson_Ti_CAS
Lesson Title: River Optimization Problem
Summary: This in an inquiry-based approach lesson to help students understand the different
methods used for optimizing (or minimizing) a particular element of a problem. Students will use
data collection, algebra, technology, and/or calculus as a means to optimize (or minimize) the
elements of a particular problem.
Key Terms: Constraint, restriction, derivative, minimum, critical points, extrema.
Background Knowledge:
Using the Pythagorean Theorem
Writing an algebraic equation
Solving an equation for an unknown
Graphing functions
Concepts of the derivative
Finding graphically: Minimums & Zeroes
NCTM & Ohio Standards Addressed:
Grade 11-12:
Standard 2: Measurement.
Benchmark D: Solve problem situations involving derived measurements; e.g., density,
acceleration.
Benchmark F: Write and solve real-world, multi-step problems involving money, elapsed
time and temperature, and verify reasonableness of solutions.
Grade 8-10:
Standard 4: Patterns, Functions and Algebra.
Benchmark D: Use algebraic representations, such as tables, graphs, expressions,
functions and inequalities, to model and solve problem situations.
Grade 8-10:
Standard 6: Mathematical Processes.
Benchmark A: Formulate a problem or mathematical model in response to a specific need
or situation, determine information required to solve the problem, choose method for
obtaining this information, and set limits for acceptable solution.
Learning Objectives:
Students will be able optimize (or minimize) appropriate details of a problem using data
collection, algebra, technology (TI Nspire), and/or calculus.
Materials: Picture, paper, grid paper, pencil, ruler, calculator (TI Nspire), link cable (optional)
Procedure: Students will work individually or in groups of two or three to complete the inquirybased handout. Each student will be responsible for their own handout, but only one will be
collected for each group.
Assessments: Instructor will informally assess student progress by walking around to each
group, listening to students explanations, and asking questions. Instructor will formally assess
student work by collecting and evaluating the written group handout.
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Part 1: Teacher’s pack
RIVER OPTIMIZATION PROBLEM
Activity 1: DATA COLLECTION METHOD
Lesson plan for teachers
The problem and the extension range in difficulty from average to challenging.
Introduce the problem to the class.
Provide each student with an instruction sheet, ruler, pencil, eraser, and a grid paper.
Prior knowledge needed:
 Measuring using rulers.
 Pythagorean Theorem.
 The distance formula (Distance traveled = rate of travel * time elapsed)
 Optimization concept (minimizing, maximizing, restrictions on the domain, writing
optimization equations, dependent variables, independent variables, data collection
method using measuring, use of tables to draw conclusions, and extrema –maximum,
minimum- concepts)
 Knowledge of scaling and converting back to the original unit.
PROBLEM:
You are standing at the edge of a slow-moving river which is one mile wide and wish to
return to your campground on the opposite side of the river. You can swim at 2 mph and
walk at 3 mph. You must first swim across the river to any point on the opposite bank.
From there, walk to the campground, which is one mile from the point directly across the
river from where you start your swim. What route will take the least amount of time?
This inquiry-based problem is to be assigned to students, after explaining the “optimization”
concept.
1. Introduction of the problem and discussion
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Draw the picture on the board using the following steps:
 Draw where you are (the route –lower straight line- and the exact point “you”).
 Draw where the campground is (the route –upper straight line- and the exact point
“campground”.
 Label the points as shown in the figure.
 Tell students that there is a slow moving river in between your location and the
campground, and write down the word “river”. Draw the vertical dotted line as shown
in the figure, and write down the width of the river (the distance between the 2 routes,
specifically the vertical distance “1 mi.”
 Ask students to find their options to return to the campground. Do not forget to have
them justify their answers. (Students can find many possible routes will be found.)
Through probing questions, lead students to realize that they should swim directly to
the campground or first swim to the opposite side, and then walk straight to the
campground.
2. Summary
While verbally summarizing, write the summary on the board.
There are two options to get back to the campground:
 Swim directly to the campground.
 First swim to get to the opposite side, and then walk straight to the campground.
Swimming to the opposite side can be done through many swimming lanes, or routes.
At this point draw one of the dotted slanted segment (CD) which represents the swimming
distance.
3. Introducing the restrictions
Tell students that in this problem, they are required to do both swimming and walking. What
are the restrictions? Lead them into answering that the walk distance cannot be 0.
Therefore 0 < walking distance  1.
3. Probing questions and students’ conjectures:
 The class should discuss how long it takes to get to the campground. Students should:
 Identify the restrictions on the walking distance
 Guess which swimming distance (straight, etc…) they think would give them the
shortest route to get back to the campground.
 Write on the board around four or five different guesses.
 Tell students that they will check their conjectures using a data collection activity.
Show them the instruction papers.
4. Review of distance formula:
Recall: if travel is at a CONSTANT rate of speed, then
Distance traveled = rate of travel * time elapsed
Write down the formula on the board.
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5. Have students solve the problem using data collection method
 Show them the instruction papers and the grid papers that they have to use to solve the
problem.
 Divide the students into groups of 4.
 Distribute the instruction papers, grid papers, rulers, pencils, erasers (all of the
materials to each of the students).
 Give students a minute to look at the instruction paper.
 Ask a student to read the directions to the class.
 Scaling and converting back to miles:
 Ask them to each reproduce the figure on their grid papers using the following
scale. Have them draw 10cm to represent “1 mile”.
 Teach them how to convert back to miles after measuring using their rulers.
 Use the following formula:
 Distance in miles = distance in cm (found by measuring using the ruler / 10, since
every 1cm represent 1 mile.)
 Have students discuss within their groups possible routes that will lead them to the
campground. Then, using the ruler, measure each of the routes that they found, and
convert their measurements to miles. They should write down their answers in the
table provided on the instruction sheet.
 Using the table, deduce which of the routes gives the shortest total time to get to the
campground.
 Have students complete the instruction sheet by answering all of the questions.
 Have students work on the extension problems.
7. Sharing answers (on the board) with the class:
 Ask each group to share two or three of their findings that another group did not share.
 Write down their answers on the board.
 Check if any of the groups have found a shorter time.
 Using the table, deduct which route gives the shortest time to get back to the
campground.
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Part 1: Student’s pack
RIVER OPTIMIZATION PROBLEM
Instruction sheet for the STUDENTS
Activity 1: Data Collection Method
Date: ___________
Class: ___________
Group #: __________
Team members: _________________________________________________________
River Optimization Problem
Solving using data collection method
The following problem is a minimum optimization problem.
Maximize your success by following these guidelines:
GUIDELINES FOR SOLVING A MAX/MIN. OPTIMIZATION PROBLEM
Using Data Collection Method
1. Read the problem slowly and carefully. It is important to know exactly what the
problem is asking. If you misread, you have no chance of solving it correctly.
2. Look at the sketch and figure out within your group what information you are given,
and what do you have to find.
3. Write down the restrictions on the domain.
4. Write down what is to be optimized.
5. Collect data and fill in the table provided.
6. Using the data collected in the table, deduce your answer.
PROBLEM:
You are standing at the edge of a slow-moving river which is one mile wide and wish to
return to your campground on the opposite side of the river. You can swim at 2 mph and
walk at 3 mph. You must first swim across the river to any point on the opposite bank.
From there walk to the campground, which is one mile from the point directly across the
river from where you start your swim. What route will take the least amount of time?
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Recall: If travel is at a CONSTANT rate of speed, then:
Distance traveled = rate of travel * time elapsed
Let “d” represent the distance traveled.
Let “r” represent the rate of travel.
Let “t” represent the time elapsed.
Then d = r * t  t = d / r
Complete the following:
a. What is the “SWIM rate? __ mph.
b. What is the “TRAVEL rate”? __ mph.
c. Recall: CD is the swim distance.
Write down the equation for the “SWIM TIME”.
_____________________________________hour(s)
Recall: DB is the walk distance.
Are there any restrictions on the walk distance? If so, what are they? Justify your answer?
________________________________________________________________________
________________________________________________________________________
Write down the equation for the “WALK TIME”.
_______________________________________________hour(s)
d. Write down in words what is to be optimized (minimized) in this problem:
_______________________________________________________________
e. Recall: total time to get back to the campground form from your location is the total of
the swim time and the walk time.
Let T = total time to get back to the campground form where you are.
Write down the equation for the total time “T”:
______________________________________________ hour(s).
Data collection method to be followed
Step 1:
Scaling: reproduce the figure on the provided grid paper using the following scale: draw 10cm to
represent “1 mile”.
Important note:
Below, you are asked to measure distances using a ruler. Remember that the answer that you get
is in “cm” according to our scaling procedure. Since in our scale every 1cm represent 1 mile, you
have to convert back your answer to miles by using the following formula:
Distance in miles = distance in cm found by measuring using the ruler / 10,
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Step 2:
- Using a ruler, measure the following distances: the swim distance “CD”, and the walk distance
“DB”.
- Recall: the answer you got is in “cm”. Thus, using the conversion formula provided in the table,
convert your answer to miles, and then record it in the table.
- Compute the swim time, walk time, and total time using the formulas provided in the table
and then Record your answer:
Recall: to convert your measurement with the ruler into miles use the following formula:
Distance to be recorded in table = Distance in inches found by measuring using ruler/10
Step 3:
Complete the table provided below.
Record your data for distance AD. Sort your answers, so that the shortest time is listed first, and
the longest time is listed last.
AD
(mile)
CD
(Swim Distance)
= measure in cm using
ruler / 10
(miles)
DB (Walk
Distance)
= measure in
cm using ruler
/ 10
(miles)
Swim Time
= swim
distance in
miles / 2
(hour/s)
Walk Time
= walk distance
in miles / 3
(hour/s)
Total Time
= Swim Time
+ Walk Time
(hour/s)
0
10 cm / 10
= 1 mph
10 cm / 10
= 1
1 / 2 = 0.5
1 / 3 = 0.33
0.83
0.1
0.2
Conclusion:
Complete the following:
a. According to the data that you recorded in the table, what is the shortest total time to
travel from where you are to the campground?
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_________________ (hour/s).
b. According to data that you recorded in the table, what value of x and which route gave
you the shortest time?
___________________________________________________________________________
___________________________________________________________________________
c. Record your data in the table provided on the board.
Extension 1:
Is there an advantage to record the distance AD in the table in an ascending order? Justify
your answer.
______________________________________________________________________________
______________________________________________________________________________
Is there any other ways order you can use to record distance AD in the table? Are there
any more advantages to using this method over the ascending order’s advantages? Give
two answers and justify each one of them.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Summarize what you have learned in this activity including formulas:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Formulas learned:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Extension 2:
Do you think that the shortest total time to travel from where you are to the campground
would be different than the one you found if you were given the choice to swim then walk,
or swim only? Justify your answer.
Answer:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
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RIVER OPTIMIZATION PROBLEM
Activity 2: Solving using ALGEBRA and Ti-CAS_Nspire
Instruction sheet for the students
Date: ___________
Class: ___________
Group #: __________
Team members: _________________________________________________________
The following problem is a minimum optimization problem.
Maximize your success by following these guidelines:
GUIDELINES FOR SOLVING A MAX/MIN. OPTIMIZATION PROBLEM
Using Algebra
1. Read the problem slowly and carefully. It is important to know exactly what the problem is
asking. If you misread, you have no chance of solving it correctly.
2. Look at the sketch and figure out within your group what do you have and what do you have
to find.
3. Write down the function to be optimized (minimized).
4. Identify the domain of the function. Knowing the restrictions is a help in this step.
5. Graph the function using your calculator and locate the minimum.
PROBLEM:
You are standing at the edge of a slow-moving river which is one mile wide, and wish to
return to your campground on the opposite side of the river. You can swim at 2 mph and
walk at 3 mph. You must first swim across the river to any point on the opposite bank.
From there walk to the campground, which is one mile from the point directly across the
river from where you start your swim. What route will take the least amount of time?
Let x be the distance denoted in the given diagram.
Recall: If travel is at a CONSTANT rate of speed, then:
(Distance traveled) = (rate of travel) (time elapsed)
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Let d represent the distance traveled.
Let r represent the rate of travel.
Let t represent the time elapsed.
Then d = rt  t = d /r
Complete the following:
a) What is the SWIM rate?
___mph.
b) What is the TRAVEL rate?
___ mph.
c) Recall: DB = WALK DISTANCE
Write down the walk distance DB in terms of x:
___________________________
Write down the walk time in terms of x:
____________________________
d) Recall: CD = SWIM DISTANCE
Write down the swim distance CD in terms of x. Show your work.
_______________________________________________________________________
_______________________________________________________________________
Write down the swim time in terms of x:
______________________________________________
e) Recall: total time elapsed is both the swim time and the walk time.
Write down the equation of the total time elapsed in terms of x:
___________________________________________________
f) Describe in words what is to be optimized (minimized):
_______________________________________________________________________
_______________________________________________________________________
g) Let T = total time elapsed.
Write down the optimization equations in terms of x.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
h) Describe the restrictions (in words) to this problem:
_________________________________________________________________________
_________________________________________________________________________
i) Write down in terms of x, the domain of the function:
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_______________________________________________________________________
_______________________________________________________________________
Now that the function to be minimized is determined, we need to graph the function using the
graphing calculator “TI_CAS_NSpire”. We also need to determine the appropriate viewing
window to observe the complete graph. Then using the minimum feature of the calculator, we
should determine the minimum for the graph on the interval from [0, 1).
Graph the function using TI_CAS_NSpire. Show the complete graph.
NOTE: for additional information on how to graph a function and find its minimum using the
TI_CAS_NSpire, refer to “Attachment A”.
What is the value of the minimum of the function?
__________________________________________
Interpret the meaning of the minimum’s coordinates of the function T(x):
__________________________________________
Conclusion:
Complete the following:
d. According to what you have just found, what is the shortest total time to travel from
where you are to the campground?
________________________________________________________________________
________________________________________________________________________
e. According to what you have just found, what value of x and which route gave you the
shortest time?
________________________________________________________________________
________________________________________________________________________
Extension:
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Do you think that the shortest total time to travel from where you are to the campground
would be different than the one you found if you were given the choice to swim then walk,
or swim only? Justify your answer.
Answer:
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Summarize the formulas you have learned in this activity:
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Remarks
Attachment A (graphing and finding the minimum)
GRAPHING THE FUNCTION: (1/ 2) 1  x 2  (1/ 3) x  (1/ 3)
Follow the steps below to graph the function:
Turn the calculator on by pressing the “on” key.
Create a new document:
Press “ctrl” then “home (the key that has the home picture on). Your screen should look like the
image below:
Press the right arrow key on the Nav pad (the big round button), your screen should look like the
image below:
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The “New Document is shaded in black; don’t change anything, Press “enter” button. Your
screen should look like the image below:
The calculator will ask you whether or not you want to save the document you were working on;
reply by “yes” if you want to save the document you were working on and on the “no” if you
don’t want to save it. You can use the arrows on the Nav pad to move between the yes and no
buttons. After you made your choice, press enter. Your screen should look like the image below:
Press the key “2” to choose the 2nd option (Add Graphs & Geometry) or use the bottom arrow
key on the Nav pad to move down to the 2nd option and then press enter. Your screen should look
like:
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Now that you have created the x and y axis, you are ready to enter the function.
You’ll see the vertical line right next to the “=” sign at the bottom of the screen blinking, this
means that you can enter the function now.
Steps to enter the function: (1/ 2) 1  x 2  (1/ 3) x  (1/ 3)
First enter the expression (1/2). Next, to enter the square root sign, press the key “ctrl,” followed
by the key “x2”. Now enter 1+x2 . Note: to enter x2, press the green key that has “x” on it, and
then the key x2.
To exit from the square root, press the right arrow on the Nav pad. Now you can type in the rest
of the expression (the part not under the radical.)
Now enter the rest of the expression as follows: -(1/3)x+(1/3). Be careful that you do not mix the
minus sign – with the negative sign (-).
Your whole expression should look like: (1/ 2) 1  x 2  (1/ 3) x  (1/ 3)
Press enter. Now your screen should look the image below:
You have just drawn the function and now you are ready to find the minimum.
Find the Minimum:
Press the menu key. Your screen should look like:
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Press the key “6” to pick “Points & Lines”. Your screen should look like
Press on the key “2”. You should go back the graph screen. Move the arrow using the Nav pad to
the graph, until you see the graph dotted and the arrow turns into a pencil with words that say
“point on”. The calculator is telling you that you picked the graph and you are ready to put the
point on it. Your screen should look like the image below:
Press “enter”, and then the escape key “esc” to get out of the option “point on”. The point on
turns into an open hand around the point that you’ve just drawn, and you’ll see the words “point
tab”. This means that the graph is also under the point. If you want to pick the graph, you can
press “tab” then “enter”; it will take you back to the graph. But don’t do this now because we
need the point and we need to move it on the graph in order to find the minimum of the function.
Notice that the calculator screen will also show you the coordinates of the point. Your screen
should look like the image below:
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Locating the minimum of the function:
Press and hold the little button that has a blue hand on it. It is located in the middle of the Nav
pad (the big round button). Keep pressing and holding until the hand closes. You just grabbed the
point; now you are ready to move the point on the graph in order to find the minimum.
Using the left arrow key, move the hand to get closer to the minimum. Your screen should look
like the image below:
Keep moving the point to the left using the left arrow, until your screen looks like the image
below:
Notice that the letter “m” appears now. You have just found the minimum of the function and its
coordinates!
Press “enter”. Your screen will look like the image below:
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Press and hold the little hand button (the little button that has a hand on it and is in the middle of
the big round button) until you see a closed hand. This means that you have grabbed the text; you
are ready to move it to another place on the screen where you can see the coordinates better.
Move the text using the arrows on the Nav pad until your screen looks like the image below:
Write down the coordinates of the minimum up to two decimal points:
Min(0.89,0.7)
End of attachment A
_____________________________________________________________________
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River Optimization Problem
Activity 3: Using the Nspire CAS to Create a Model to Minimize Travel Time
USING CAS TO MODEL OPTIMIZATION
Instruction sheet for the Students
Date: ___________
Class: ___________
Group #: __________
Team members: _________________________________________________________
Materials: TI_Nspire_CAS, and link cable for possible screen captures
Goals: To use the TI Nspire’s construction capabilities to create a model that will allow students
to effectively determine the minimum travel time.
Problem: You are standing at the edge of a slow-moving river which is one mile wide and wish
to return to your campground on the opposite side of the river. You can swim at 2 mph and
walk at 3 mph. You must first swim across the river to any point on the opposite bank. From
there walk to the campground, which is one mile from the point directly across the river from
where you start your swim. What route will take the least amount of time?
Create a model using TI_NSpire_CAS:
1) Open a new document by pressing Home 5: New Document or ctrl N (Tool, 1: File, 1: New
Document). Hide the axes and entry line (Menu 2: View, 1: Hide Axes and 3: Hide Entry
Line).
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2) Show the scale. (Menu 2: View and 7: Show Scale) Move the pointer over the scale icon in
the upper right hand portion of the graph window, click once, and enter .1 mi in place of 1 cm.
Then press “enter”.
3) Construct a segment on the screen, labeling the endpoints A and B immediately after
constructing them (Menu 7: Points & Lines, 5: Segment).
4) Measure segment AB (Menu 6: Measurement, 1: Length) and drag an endpoint of the
segment until the length is 1 mi.
5) Draw a line perpendicular to AB through A (Menu 9: Construction, 1: Perpendicular).
6) Draw the segment AC on the perpendicular line (Menu 7: Points & Lines, 5: Segments).
Measure the length of AC, and adjust the length by dragging an endpoint so that it has a length
of 1 mi (Menu 6: Measurement, 1: Length).
7) Hide the perpindular line (Menu 2: Hide/Show, click on the perpendicular line).
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8) Place a point D anywhere on AB (Menu 6: Points & Lines, 2: Point On). Create the segment
CD and the segment DB (Menu 6: Points & Lines, 5: Segment).
You can use the attributes to change the style of segments CD and DB. Try making them
dashed and bold (Menu 1: Actions, 3: Attributes).
9) Calculate the lengths of segment CD and DB (Menu 7: Measurement, 1: Length). Drag the
measurements to an open area on the screen, and use the text tool to label the measurements
CD and DB respectively (Menu 5: Text, click next to measurement, input letters, press
enter).
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10) Grab point D, and drag it along segment AB (place the open hand over point D, press and
hold center button till hand closes). What happens to the values for CD and DB?
___________________________________________________________________________
___________________________________________________________________________
11) Is it clear that CD represents the possible swimming distance and DB represents the possible
walking distance for the problem? __________________________
12) In order to set up the data in list form, we have to first store the value that is to be collected.
First select the value for segment CD, and press sto (ctrl, var). Replace var with SD, for
swimming distance, and press enter twice. Repeat this process for the value of DB; replace
var for WD, walking distance.
13) Press Home, 3: Lists & Spreadsheets. This will open up a blank spreadsheet for you to
collect your data on walking/swimming distance, walking/swimming time, and total travel
time.
14) Place the pointer in the white space to the right of the shaded title of the column (A), and
enter S as the name. Repeat for the other columns (B-E). Enter W (walking distance) for
column B, ST (swim time) for column C, WT (walk time) for column D, and TT (travel
time) for Column E.
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15) Hold ctrl and press left in order to return to the graph window. Now select and drag point
D so that it lies on point A. This will give us a starting point for when we start recording the
data. Hold ctrl and press right to return to the spreadsheet.
16) Next, place your cursor in the formula cell just below the column name, start with the
swimming distance column (s). Press Menu 3: Data, 2: Data Capture, 1: Automated Data
Capture, and press the var key and select the variable labeled sd. Repeat the steps for the
walking distance column, selecting the variable wd after pressing the var key.
If you moved point D to lie upon point A, then the data in both columns should be 1. What
do these values represent in terms of the problem?
___________________________________________________________________________
___________________________________________________________________________
17) Move your cursor into the formula cell just below the column name for swimming time (st).
In the formula cell enter s/2 and press enter. The conflict detected screen may emerge, if so
choose from the drop down box variable reference and select okay. Your value in the first
row should be half the value of swimming distance in the same row.
Repeat these steps for the walking time (wt) column, entering w/3 into the formula cell.
Your value should be one third the walking distance value.
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18) Why did we divide the swimming distance by 2? Why did we divide the walking distance by
3?
___________________________________________________________________________
___________________________________________________________________________
19) Place your cursor in the formula cell for the column for travel time (tt). Enter st + wt, and
press enter. Observe the value presented.
In terms of walking distance and swimming distance, what is the equation for the individuals
total travel time?
___________________________________________________________________________
___________________________________________________________________________
20) Hold ctrl and press left to return to the construction page.
21) Take a moment to hypothesize about what you believe the shortest route will be. Make an
educated guess as to the best walking and swimming distance that you believe will return the
shortest travel time.
___________________________________________________________________________
___________________________________________________________________________
22) Use your pointer to grab point D. (Position the open hand next to point D and hold the
center button until the hand closes).
Drag point D along segment AB. Remember the speed at which it is drag dictates how much
data is captured in the spread sheet. Try dragging slowly. Drag until the point D lies on top
of point B, or until the WD (walking distance) value = 0 mi.
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23) Hit escape to exit the drag feature. Hold ctrl and press right on the directional pad to return
to the spreadsheet window.
Was your data from dragging point D collected?
___________________________________________________________________
24) Examine your data. What is the smallest value recorded in the travel time column (tt)?
__________________________________________________________________
What does this value mean in terms of the optimization problem?
___________________________________________________________________________
___________________________________________________________________________
25) With the minimal travel time determined, what is the optimal swim distance to minimize
travel time? _____________________________________
What is the optimal walk distance to minimize travel time? ________________________
26) How close was your hypothesis in question 22?
___________________________________________________________________________
___________________________________________________________________________
27) Put all of the information together and restate in words what route will take the least amount
of time, and how much time that route will take.
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___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Further Discovery
28) Return to the Spreadsheet page. In column F, enter “x” into the label or name box next to the
shaded box. In the formula cell enter 1-w, press “enter”.
Is it clear that 1-w represents the length of segment AD? The maximum walk distance is one
mile, so x will represent the distance not walked, or one mile minus the miles actually
walked.
29) Open a Data & Statistics page. (Home, 2: Graphs & Geometry)
30) Create a scatter plot, using “x” as your x variable and “tt” as your y variable. (Menu, 3:
Graph Type, 4: Scatter Plot)
31) Adjust your graph window to better fit the scatter plot. (Menu, 4:Window, 9:Zoom-Data).
32) Examine the data plot. Based on the shape of the plot, what regression model do you
hypothesize as being the best fit? Why?
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___________________________________________________________________________
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Dr. Antonio R. Quesada – Director, Project AMP
33) Open a Calculator page. (Home, 1: Calculator).
34) Calculate different regression models for the previous data and choose the best fit. Use “x”
as your X list and “tt” as your Y list. (Menu, 7: Statistics, 1: Stat Calculations, choose a
model)
What is the best regression model for this data? Why? (Hint: compare the data plot with the
actual regression line)
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35) What is the equation for the regression model?
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36) Return to the Graph Page. Place a point any where on the regression line.
37) Grab and drag the point along the regression line until a small box with the letter “m”
appears inside. This is the minimum for your regression line.
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Dr. Antonio R. Quesada – Director, Project AMP
What is the coordinate of the minimum for the regression model? _______________
38) What is the relationship between the y coordinate from the minimum and the problem?
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What is the relationship between the x coordinate and the original problem?
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39) Compare the coordinates found for the extrema of the regression model to the data collected
in your spreadsheet, are the results for the minimum travel time consistent? Determine the
swimming and walking distance for both and compare, are they similar?
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EXTENSION
I) In a previous activity the equation (1/ 2) 1  x 2  (1/ 3) x  (1/ 3) was given as the equation to
determine total travel time for the optimization problem. Use your graphing calculator to
compare this equation to the equation of the regression model discovered in question 35.
→
II) Are the graphs the same? _________
III) If the graphs are not the same, why is the regression model still a good fit for the data
collected? (Hint: keep in mind the constraints on the problem)
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IV) If the problem was changed to incorporate 10 miles of shore line and a 5 mile wide river,
would the regression model still fit? Would it still match up well with the actual equation?
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___________________________________________________________________________
___________________________________________________________________________
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Dr. Antonio R. Quesada – Director, Project AMP
River optimization problem
USING CALCULUS FOR OPTIMIZATION
Instruction sheet for the Students
Activity 4: Determining the quickest route using Calculus
Materials: TI-Nspire CAS, and link cable for possible screen capture
Goals: Using calculus to minimize travel time, through the calculation of the first derivative and
the discovery of critical points.
Problem: You are standing at the edge of a slow-moving river which is one mile wide and wish
to return to your campground on the opposite side of the river. You can swim at two mph and
walk at three mph. You must first swim across the river to any point on the opposite bank.
From there you must walk to the campground, which is one mile from the point directly
across the river from where you start your swim. What route will take the least amount of
time?
Recall: If travel is at a CONSTANT rate of speed, then:
(Distance traveled) = (rate of travel) (time elapsed)
Let d represent the distance traveled.
Let r represent the rate of travel.
Let t represent the time elapsed.
Then d = rt  t = d / r
Complete the following:
1) Looking at the problem, describe in words what is to be optimized (minimized)?
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2) Describe in words the restrictions on the walking distance in this problem?
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Dr. Antonio R. Quesada – Director, Project AMP
___________________________________________________________________________
___________________________________________________________________________
3) What is the walk rate?
_______________________
4) What is the swim rate?
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5) The distance from A to B is 1 mile. Write down an equation in terms of x for the walking
distance, given D can fall anywhere on AB:
__________________________
6) The direct route across the river from point C is perpendicular to AB. Given that angle DAC
is a right angle, use the Pythagorean Theorem to determine the equation for the length of
CD in terms of x:
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7) CD is the swim distance, write down the equation for the swim time in terms of x:
________________________________________________________
8) DB is the walk distance, write down the equation for the walk time in terms of x:
_________________________________________________________
9) Recall: Total time elapsed is the swim time plus the walk time.
Write the equation for calculating the total travel time in terms of x.
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10) Now that we have determined the function to be minimized, we need to graph the function
using the graphing calculator “TI_CAS_NSpire”.
GRAPHING THE FUNCTION
11) Start off by opening a new Graphing and Geometry page on the Nspire.
12) Move your pointer down to the function entry line, and click once. Under f1(x), enter the
function you discovered in Question Six for minimizing the travel time. Graph the function.
(Travel Time = (1/ 2) 1  x 2  (1/ 3) x  (1/ 3) )
NOTE: for more information on how to graph the function, refer to “Attachment A”
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Dr. Antonio R. Quesada – Director, Project AMP
13) After graphing the function, hide it using the “hide” button next to the f1(x) in the equation
entry line.
14) Put your cursor back into the equation entry line. This time you will be entering the
equation to find the first derivative. Enter
into the line for f2(x) and press “enter”
to graph it. (Press ctrl × to access the template list, and choose the “derivative”
template).
15) The first derivative can be used to determine the critical points of a function. Critical
points occur when the derivative (or slope of the line when visualized) is equal to zero or a
point where the function ceases to be differentiable. Critical points can be used to indicate
the local minimums and local maximums of a function.
In order to determine those critical points you have to set the equation of the first derivative
to zero, or determine the zeroes of the graph of the first derivative.
16) Does the graph of the first derivative have any zeroes?
__________________________________________________
Is the zero a minimum or a maximum, and how can you tell? (Hint: the graph of the
derivative illustrates slope, so if the graph is below the x axis, the slope is negative at that
point, and visa-versa)
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Dr. Antonio R. Quesada – Director, Project AMP
__________________________________________________________________________
__________________________________________________________________________
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17) Change the window settings to x-min = 0, x-max= 1.5, y-min = -1, and y-max = 1.
18) Place a point on the graph of the first derivative. Use the pointer/hand to grab that point and
drag it towards the location where the graph crosses the x axis, the zero. Keep dragging the
point until a small box with a zero inside it appears next to the hand. This is your zero.
19) What is the location of the zero for the graph of the second derivative?
__________________________________________________________________
20) Check this value by opening a Calculator page, and entering
Press “answer”. Are the answers the same?
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Dr. Antonio R. Quesada – Director, Project AMP
.
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21) Now that we have determined the x value at which the minimum of the function occurs, we
can now determine the swimming distance and walking distance that coordinate to the
minimal travel time.
What is the minimum total travel time? (Hint f (.894427) =?) _______________________
What is the swimming distance that results in the minimum travel time?
___________________________________________________
What is the walking distance that results in the minimum travel time?
____________________________________________________
22) Express your answers to Question 20 in words: “What route will take the least amount of
time?”
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Journal Activity
I) List any formulas/properties/definitions that you have discovered from this activity.
II) Outline the process for finding critical points using the first derivative.
END OF STUDENT’S EDITION: ACITIVITIES
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Dr. Antonio R. Quesada – Director, Project AMP