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PROBLEM-SOLVING with the TI-NSPIRE CAS CALCULATOR
ICTMT 9Metz 2009
PresentersPauline Holland
Shirly Griffith
Background
• Education in Australia is state based• Victorian Curriculum and Assessment Authority
(VCAA) set the examinations for the State of Victoria• 2002 - VCAA implemented a pilot mathematics subject
in a small number of schools, called Mathematical Methods CAS.
• 2006 - The subject Mathematical Methods CAS was open to all students wishing to study calculus based mathematics
• 2009 - Mathematical Methods CAS is a prerequisite for most science based university courses.
• From 2010, students undertaking Specialist Mathematics will be assumed to be using CAS technology.
• Mathematical Methods CAS and Specialist Mathematics are assessed as follows:
1. School Assessed Coursework (34%)
2. Examination 1 Technology Free (22%)
3. Examination 2 Technology Active (44%)
Getting Started
• When the TI-Nspire CAS is first turned on, it starts with the HOME c screen as shown.
• You can return to this screen at any time by pressing the HOME c icon.
• The System Info folder contains tools that will allow the user to change the settings on the calculator.
Checking the Operating SystemTo check which operating
system is on your calculator,
press:
• 8: System Info 8• 5: About 5The calculator being used
here has OS 1.7.2733
installed.
To select OK, press:
• Tab e• Enter ·
System Settings
To change System
settings, press:
• HOME c• 8: System Info 8• 3: Graphs & Geometry
3Use Tab e to move
between fields. Press
Click x to select options.
Question 1
A debt of $450 is to be shared
equally among the members
of the Crackerjack club. When
five of the members refuse to
pay, each of the other
members will have to pay an
additional $3. How many
members does the
Crackerjack club have?
Solution
Let the number of members
be n.
Let the value of the individual
debt be s.
So 450 450and 3
5s s
n n
On a Calculator page,
press:
• Menu b• 3: Algebra 3• 1: Solve 1Complete the entry line as:
Then press Enter ·.
450 450solve 3
5n n
The Crackerjack club has 30
members.
Check
If all members paid, then each
member would pay
.
If only 25 members paid then
each paying member will have
to pay an additional $3.
i.e. $18
450$15
30
Question 2
Instead of walking along 2 sides of a rectangular
field, Patrick took a short cut along the
diagonal, thus saving distance equal to half the
length of the shorter side. Find the length of the
long side of the field given that the length of
the short side is x metres.
Solution
Let the length of the diagonal
be d metres.
Let the length of the longer
side be y metres.
Given the length of the shorter
side is x metres.
2 2d x y
1
2d x y x
Complete the entry line
as:
Then press Enter ·
The longer side is equal to
metres.
2 2 1solve ,
2x y x y x y
3
4
x
Question 3
Suppose that the annual dues of a union are as
follows:
If d is the annual dues and s is the salary, graph the relationship
and determine the annual union dues of an employee earning
$62 000.
Employee’s Annual Salary Annual Dues
Less than $40 000 $400
$40 000 or more but less than $60 000
$400 + 1% of the salary in excess of $40 000
$60 000 or more $600 + 2% of the salary in excess of $60 000
Solution
The relation between the annual dues d and the
salary s can be expressed as:
400 if 40000
( ) 400 0.01( 40000) if 40000 60000
600 0.02( 60000) if 60000
s
d s s s
s s
To graph this hybrid function
On a Calculator page, select
the hybrid function template
from the Maths expression
template by pressing
• Ctrl /• × rUse the NavPad to move
across to the selection
highlighted.
Then press Enter and
select 3 function pieces.
Press Enter ·.Type the expressions as
shown.
Tab e to move between
fields.
Do not press Enter ·.
To define the function,
press:
• Tab e• Ctrl /• Var h
Type f1(x), then press
Enter ·.
To draw the graph of the
hybrid function, open a
Graphs & Geometry page,
press:
• Menu b• 4: Window 4• 1: Window Settings 1Complete the table as
shown.
Press Enter ·.To determine the annual union
dues of an employee earning
$62 000, press:
• MENU b• 6: Points & Lines 6• 1: Point 1Move the cursor to the line and
when a point appears press
Click x twice.
Use the NavPad to move the
cursor over the x-coordinate
of the point, when an open
hand appears, press Click
twice and change the x-
coordinate to 62 000.
Press Enter ·.The annual dues of an
employee earning $62 000 is
$640.
Question 4
Lauren and Matt are travelling off road in the desert in Morocco. They are 10 km from a long, straight road. On the road, their 4WD can do 60 km/hr, but off road, it can only manage 40 km/hr. It is getting dark and Lauren and Matt are keen to reach
the town where they are staying for the night. The town is 25 km down the road (from the nearest point P on the road).
(a) How many minutes will it take for Lauren and Matt to drive to the town through the desert?
(b)Would it be faster if they first drove to P and then used the road to town (T)?
(c) Find an even faster route for Lauren and Matt to follow.
Draw a diagram
25
10
ROAD
DESERT
T P
A
Solution
(a)
40 km/hr = ⅔ km/min
Time taken =
2 2d( ) 10 25 5 29 29.93kmAT
35 29 40.39min
2
10 km at ⅔ km/min will take 15 minutes.
25 km at 1 km/min will take 25 minutes.
A total time of 40 minutes.
This would save 0.39 minutes so it would be faster if Lauren
and Matt drove to the road and then to the town.
c To find a faster route, chose a point M, x metres from P on the
road and construct a function F(x) for the total time it would take for Lauren and Matt to drive to M and then along the road T.
Find the value of x which would make F a minimum.
25
10
ROAD
DESERT
T Mx
P
A
Time taken = minutes
Time taken = (25 − x) minutes
The total time taken is:
2 23( ) 10 (25 )
2F x x x
( ) (25 )d MP x
2 2310
2x
2 2( ) 10d AM x
A graphical method of finding
a solution
On a Graphs & Geometry
page, complete the
Function entry line as:
Then press Enter ·.Adjust the Window Settings as
shown. (the maximum x could
be is 25 and we know that the
total time, y, is under one hour
or 60 minutes)
2 231( ) 10 (25 )
2f x x x
To locate the minimum, press:
• Menu b• 5: Trace 5• 1: Graph Trace 1Move the cursor along the
curve until the minimum is
located or Press ? for hints and
then n for minimum.
The minimum time is 36.2
minutes when x is 8.94 km.
An algebraic method of finding a solution
It is possible to find the
minimum time taken by using
calculus.
Find x when f’(x) = 0
On a Calculator page, complete
the entry line as:
Then press Enter ·.The minimum time is 36.18
mins, when x is 8.94 km
dsolve( 1( ) 0, )
df x x
x
Question 5
A confectionery manufacturer
wishes to market the latest
chocolate sensation in an
eye-catching pyramid shape
package. The volume of the
pyramid is to be 1000 cm3
and the base must be a square.
Find the dimensions of the
packaging if the
manufacturer wishes to keep
the surface area to a
minimum.
Solution
Volume = ⅓x2h = 1000
To find the perpendicular
height of a sloping side,
complete the entry line as:
Then press Enter ·.
2
3000h
x
2 2
2
3000
2
x
x
The surface area of the
pyramid is made up of the
square base and four triangular
sloping sides.
On a Calculator page, define
SA as f1(x).
62
2
1 360000004
2 2
xSA x x
x
To find the value of x, when
the surface area is a minimum,
complete the entry line as:
Then press Enter ·.To find the height , complete
the entry line as:
Then press Enter ·.
dsolve( ( 1( )) 0, )
df x x
x
2
3000
12.849
The surface area will be a minimum when the base is 12.85 cm and the height is 18.17cm.
Question 6
Two Year 7 classes complete the same end-of-year mathematics
test. The marks expressed as percentages are given in the
following table.
Is it possible to determine which class overall has achieved at a
higher level?
7A 40 45 50 52 54 57 60 68 63 75 80 89 85 90
7B 60 63 70 74 77 82 80 79 81 87 73 90 95 97
To compare the two classes, use parallel boxplots.
On a Lists & spreadsheets
page, enter the data for 7A into
Column A and label the
column ClassA, the data for
7B into Column B and label
the column ClassB.
To highlight Column A, with
the cursor in the top cell, press
the up arrow £ on the
NavPad, then press:
• Menub• 3: Data 3• 6: Quick Graph 6• Menu b• 1: Plot Type 1• 2: Box Plot 2Repeat for Class B
The scales are different.
To change the scale for Class B, press:
• Ctrl /• Menu b• 5: Zoom 5• 1: Window Settings 1Change the Window Settings
to match the scale for Class B.
It is now possible to compare
the results of the two classes.
Class B has a smaller range
and IQR. Class B has a
substantially higher minimum,
medium and maximum. Class
B appears to have achieved
better results overall than
Class A.
Question 7
The following beep test data was obtained from a Year 11
Physical Education student. The table below shows the heart rate
recorded each minute. Use this data to predict the heart rate of a
student at 6.5 minutes.
Time 0 1 2 3 4 5 6 7 8 9 10 11 12
Heart Rate
78 137 150 157 170 179 183 190 194 197 203 207 208
To draw a scatterplot of the
data, on a Lists &
Spreadsheets page, enter the
data in Columns A and B,
labelling as shown.
Then press:
• HOME c• 5: Data & Statistics 5Press Tab to enter the variables
on the axes.
The data could possibly be
logarithmic in shape.
To fit a logarithmic regression
line to the data, return to the
Lists & Spreadsheets page and
delete the first row.
( loge0 is not possible) then
press:
• Menu b• 4: Analyze 4• 6: Regression 6• 9: Show Logarithmic 9The regression line and it’s
equation will be displayed.
To predict the heart rate at 6.5
mins, return to the Lists &
Spreadsheets page, press:
• Menu b• 4: Statistics 4• 1: Stat Calculations 1
• B: Logarithmic Regression B
Complete the table as shown.
Then select OK.
Note: The equation has been
saved as f1(x).
To predict the heart rate, on a
Calculator page, complete the
entry line as:
f1(6.5)
Then press Enter ·.Using the logarithmic
regression model, at a time of
6.5 mins the heart rate is
predicted to be 187.55.
Question 8
John rides a Ferris wheel for five minutes. The diameter of the
wheel is 10 metres, and its centre is 6 metres above the ground.
Each revolution of the wheel takes 30 seconds. Being more than
9 metres above the ground causes John to suffer an anxiety
attack. For how many seconds does John feel uncomfortable?
10 m
9 m
Anxiety
6 m
• Maximum height: 11 m
• Minimum height: 1 m
• Amplitude: 5
• Period: 30 s
An initial model of this
situation might be:
To graph this function, on a
Graphs & Geometry page
complete the entry line as:
( ) 5sin 615
xf x
1( ) 5 615
xf x
Clearly this function is not
quite right as it suggests that at
the start of the ride, f(0) , John
was 6 m off the ground.
John would start the ride when
the chair is at the lowest point.
That is when it is 1 m off the
ground. To discover when the
model above is at the lowest
point, press:
• Menu b• 5: Trace 5• 1: Graph Trace 1
Use the NavPad to trace to the minimum.The minimum point will be displayed on the graph screen. It is (22.5, 1).
Use this information to modify
the equation that models
John’s situation.
Sketch this graph.
1( ) 5sin ( 22.5) 615
f x x
John is uncomfortable when
the Ferris wheel is more than 9
metres above the ground. Complete the entry line as:
f2(x) = 9
To find the points of
intersection, press:
• Menu b• 6: Points & Lines 6• 3: Intersection Point(s) 3Move the cursor to a point of
intersection and press Click x twice.
For one rotation, John will feel uncomfortable for 19.43 − 10.57 = 8.86 secs.
Question 9After Adam finishes his
Multimedia Design course, his
first job pays him a weekly
salary of $600 after tax. He sees
a second hand Peugeot for sale at
$35 000. He has a deposit of
$2000 and can get a personal loan
of $33 000 at 13% p.a.
compounded monthly for 7 years.
a. How much money will Adam have to live on each week, after he makes a monthly repayment?
b. How much money will he still owe after 5years?
Solution
To calculate the monthly
repayments, on a
Calculator page, press:
• Menu b• 8: Finance 8• 1: Finance Solver 1Complete the table as shown.
Return to the Pmt field and
press Enter ·.
The monthly repayment will be $600.33
The yearly repayments will be: 600.33 x 12= $7203.96
So weekly repayment will be $138.54
To determine how much
money he still owes after 5
years (60 months), on a Graphs
& Geometry page, press:
• Menu
• 3: Graph Type
• 1: Parametric
Complete the entry lines as
shown.
Use Trace to see the balance.
After 5 years, $12, 627.92 is
still owing.
Question 10
Tasmania Jones’ wheat field lies between two roads as shown in the diagram below. Main road lies along the x-axis and Side road lies along the curve with equation 3 x xy e e
a. The y- intercept of the graph representing Side Road is b. Show that b = 1.
1 mark
Solution When x = 0,
Therefore b = 1.
0 03 1y e e
b. Find the exact value of a. 1 mark
Since a is the intersection
of the Side Rd with the x-
axis, on a Calculator
page, press:
• Menu b• 1: Actions 1• 1: Define 1Complete the entry line as:
Define
Press Enter ·.
( ) 3 x xf x e e
c. c. Since a is close to 1, Tasmania finds an approximation to the area of the wheat field by using rectangles of width 0.5 km, as
shown on the following diagram.
i Complete the table of values for y, where , giving values correct to two decimal places. 1 mark
3 x xy e e
x -0.5 0 0.5
y
On a Calculator page, complete the entry lines as shown.
x -0.5 0 0.5
y 0.74 1 0.74
ii Use the table to find Tasmania’s approximation to the area of the wheat field, measured in square kilometres, correct to one decimal place. 2 marks
Area = 0.5(f(-0.5)+f(0) +
f(0.5))
Area = 1.2 km2
Complete the entry line as:
solve(f(x) = 0,x)
Then press Enter ·.
5 3log
2ea
iii. Tasmania uses this approximation to the area to estimate the value of the wheat in his field at harvest time. He estimates that he will obtain w kg of wheat from each square kilometre of field. The current price paid to growers is $m per kg of wheat. Write a formula for his estimated value, $V, of the wheat in his field. 1 mark
Area of field is 1.2 km2.
$V = m × w × 1.2
$V = 1.2mw
d. Tasmania Jones decides to find to another approximation to the area of the wheat field. He approximates the curve representing Side Road with a parabola which passes through the points (0, 1), (1, 0) and (−1, 0). He finds the area enclosed by the parabola and the x-axis as an approximation to the area of his wheat field.
i. Find the equation of this parabola 1 mark
On a Calculator page,
complete the entry line
as:
Define h(x) = ax2 + bx + c
Press Enter ·.Complete the entry line as:
solve(h(0) = 1 and h(1) = 0 and
h(-1) = 0, a)
The equation of the parabola
is:
y = –x2 + 1
ii. Find the area enclosed by the parabola and the x-axis, giving your answer correct to two decimal places. 2 marks
To find the area enclosed by
the parabola and the x-axis:
Area =
Area = 1.33 km2.
1 2
02 ( 1) dx x
iii. Find the values of k, where k is a positive real number, for which the equation 3 – kex – e –x = 0 has one or more solutions for x. 4 marks
Note the Calculator screen
opposite.
The calculator does not return
a useful answer.
It is necessary to solve this by
hand.
When 3 – kex – e –x = 0 let u = ex
ku2 –3u+1=0 for one or
more solutions,
k is a positive real number, so,
13 0ku
u
9 4 0k
90
4k
Thank you for your attention.
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