Maths Task- Rates of Change, Probability, Logger Pro task


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Week 11 Page 2
PART 1: Rates Of change
1. A tap fills an eighty litre tank in 5 minutes. What is the rate of flow of water with respect to
time, in litres per second?
Sketch the position-time graph for x(t) = 4t – t2, 0 ? t ? 5 .Show ALL working.
Find the gradient at: (i) t = 0
(ii) t = 2
(Hint: Use Method 2 on page 6)
Hence, give the instantaneous rate of change of position with respect to time
(that is, velocity) at (i) t = 0
(ii) t = 2
(iii) t = 3
(iv) t = 4
Sketch the velocity – time graph from t = 0 to t = 5
t (sec)
(iv) t = 4
Draw the velocity-time graphs for the position-time graphs below:
428605– 245
(iii) t = 3
t (sec)
the position-time graph for the velocity-time-3graph below.
that at time t = 0, the position is 0 metres.)
-5 v (m)
8 t (sec)
A car travels at a speed of 20 km/hour for 5 minutes and at a speed of 60 km/hour for 30
minutes. Find the average speed (in km/hr) of the car for the whole journey.
CAS Calculator
The displacement of a particle is given by s = t3- 6t2 + 9t- 4, 0 ? t ? 5
Plot this function on your CAS calculator, and copy it accurately onto your
work for submission.
(b) At what times is the velocity zero?
For what values of t is the velocity negative?
Week 11 Page 3
PART 2: Probability
List the sample space for each of the following experiments:
A die is rolled
A coin is tossed
A coin is tossed and a die is rolled
Represent all possible outcomes for (c) in a lattice diagram.
Two letters from the word DOG are chosen. The same letter can be used twice.
Show all possible outcomes on a tree diagram
Calculate the probability that the letter O is chosen first and the letter G is
chosen second.
When twodice are rolled what is the probability that:
both numbers are the same?
both numbers are fives?
the total is less than four?
In each case, include in your answer a lattice diagram and highlight the possible
80 students attended a scout’s camp where surfing was offered in the morning
and bushwalking in the afternoon. Every student attended at least one activity.
44 students went surfing and 60 students went bushwalking.
(i) Use a Karnaugh map to represent this information.
(ii) Use the Karnaugh map to find the probability that a student chosen at
(a) Did not undertake either of these activities
(b) Participated in both activities
(c) Went surfing, but not bushwalking
There are 8 chocolate and 4 strawberry ice-creams in a box. Show all possible
outcomes on a tree diagram, if two ice-creams are taken out of the box
simultaneously and without looking.
Find probability that 2 ice-creams of different flavour have been taken out.
Week 11 Page 4
One hundred students were asked their preference of soft drink: Coke or Pepsi?
Twenty-five of the forty girls preferred Pepsi and forty of the boys preferred
Illustrate this information on a Karnaugh map.
Find the probability that a randomly selected student:
Is a girl
(ii) Prefers Pepsi
(iii) Is a boy who prefers Coke?
(b) If the student selected is a girl, what is the probability she prefers Pepsi?
7. The new chocolate shop specialises in all kinds of chocolates; however, it is
becoming famous for its milkshakes; namely, vanilla, strawberry and chocolate.
A survey of 100 people was conducted to determine the most popular flavours.
On top of the list is 45 people like chocolate milkshake, 40 like vanilla, 20 like
vanilla and chocolate, 11 like all three flavours, 14 like vanilla and strawberry,
17 like exactly two flavours and 26 like only strawberry. Represent this
information on a Venn diagram and use this to find the probability that a person
(a) Likes only chocolate milkshake
(b) Likes strawberry
(c) Likes chocolate and strawberry, but not vanilla
(d) Only likes one flavour of milkshake
Week 11 Page 5
PART 3: Modelling Task: Logger Pro
SEND: Modelling Tasks (Graded)
The work for submission this week is an assessed task (graded A-E) that requires you to
demonstrate your understanding of the concepts and skills, including competence with the CAS
calculator, for the topics studied in weeks 5, 7, 8 and 9.
Full marks will be awarded to students who show all their working out including the
calculator screens for the equations and graphs where specified in the tasks.
Question 1.
Modelling a path around the lakes in Happy Valley Springs.
The committee for the National Parks and Gardens has decided that a bush track around the
lakes in Happy Valley Springs would encourage more people to the park and has called for
submissions from tenders. The proposed path for the bush track around the two distinct groups
of lakes needs to satisfy the following specifications: It must follow a path from point A to point
Cand go through the lookout point at B, as shown in the diagram below.
Your task is to write a report for the National Park authority with all relevant information related
to the equation of the path that needs to be constructed, consistent with their specifications.
From the diagram above, the coordinates of the points are: A (0, 1), B (5, 2.5) and C (10,
5.5)and the distances are in kilometres.
1. The path from A to C will be a hybrid functionthat consists of two cubic functions; one to
describe the path from point A to B and the other for the path from point B to C.
Path from A to B
Week 11 Page 6
For the first part of the path, you need to locate two coordinates X and Y between points A and
B that you think satisfy the specifications. Use your CAS calculator and refer to the
methoddescribed in Week 8 to determine the equation of a cubic function for the first part of the
hybrid function.
Draw in and label the two points X and Y that you selected and accurately sketch your graph on
the grid on page 8.
2. Path from B to C
For the second part of the path you start at point B and finish at point C. Here you will use the
‘turning point form’ of a cubic equation, that is, y = a(x – b)3 + c to determine the second
cubic function of the hybrid function, if B (5, 2.5) is the point of inflection for the second cubic
Lightly sketch this function on the grid on page 8. You should find that this cubic function
comes dangerously close to one of the lakes. Explain why this is so. Show your calculations.
Because of this close distance to the lake, this function cannot be a suitable track.
We therefore need a new function to replace the cubic function from point B to point C.
Consider a quartic function to describe a suitable path from B to C, being mindful that the path
must satisfy the specifications that it must be within the dotted region, can pass between the
lakes, and must not be within each of the two groups of lakes.
3. Alternative path from B to C
The equation of a quartic function from B to C can be done in either of two ways.
a) Select additional points and use a similar method to the one you used to determine the
cubic equation for the path from A to B. However, this time you will be finding the
equation of a quartic function.
b) Use the turning point form of a quartic function, y = a(x – b)4 + c, to determine the path.
Using B as the turning point.
Draw the graph of your quartic function on the grid on page 8, showing all relevant features,
including the coordinates of additional points you used.
The following elements need to be evident in your final report for the bush track in
Question 1.
A brief statement (abstract) of the contents of your report.
All working out needs to be shown; algebraic calculations and calculator screen
captures, together with full explanations.
The equations of the two hybrid functions, remembering that the first hybrid
function consists of :
(a) An equation for the cubicfunction for the path from A to B.
(b) An equation for the initial cubicfunction for the path from B to C.
The second hybrid function consists of:
Week 11 Page 7
(a) An equation for the cubicfunction for the path from A to B.
(b) An equation for the quartic function for the path from B to C.
In addition, you are required to state the appropriate domains with both hybrid
Tear out the grid on page 8 on which you clearly show the points you have selected
between points A and C around the two distinct groups of lakes and graphs of the
functions that describe your path from A to C that are clearly drawn and labelled.
A conclusion, stating what you have learned and any suggestions you may have that
the National Parks and Gardens may like to consider in their planning of the bush
Question 2. Analysis of travel graphs.
Two bike riders set out on a 2 hour bike trip.
The following graph is the displacement-time graph of bike rider A during the trip
s (km)
t (h)
Use the displacement-time graph above to draw a velocity-time graph
What was the rider’s velocity during the first hour?
During what time(s) did the rider rest?
What was rider’s velocity in the last 0.4 of an hour?
Describe the rider’s trip, including key features such as the distances travelled and the
velocity at various stages during the trip.
Week 11 Page 8
The following graph is the velocity-time graph of bike rider B during the trip
v (km/h)
t (h)
Describe the rider’s trip, including key features such as the distances travelled and the
velocity at various stages during the trip.
The area under a velocity-time graph is the displacement.
Use the velocity-time graph above to sketch a displacement-time graph.
What was the total distance travelled?
At what time(s) did the rider rest?
Calculate the average velocity that the rider travelled on the return trip (from 1.2 hours).
Question3. Data Logging Activity
(This is optional, you obviously can’t access the example videos, but choose
your own, that sounds similar in Logger Pro)
DECV has a licence for a program called Logger Pro. In this question, you are asked to use the
Logger Pro programon your home computer to complete a report on time, distance and velocity
data gathered from a few experiments. The files for this data are located in the Distance
Velocity folder on your course CD (under Further Resources).
The activity will involve data collected using a motion sensor. You can access Logger Pro from
the Toolbox on the online site. You should have done this previously for your work in week 7.
The files in the Distance Velocity folder contain time, distance and velocity data as well as
videos of the experiments. Your task will be to analyse this data and calculate displacement and
velocity values from one of these experiments.
You are expected to view the videos and the data from the experiments and to write a report on
your analysis. More detailed instructions are given on the next page as well as in the Distance
Velocity folder (on your course CD.)
Week 11 Page 9
You are encouraged to complete this activity.The collection of real data will give you a greater
understanding of distance-time and velocity-time graphs. If you have difficult accessing the
relevant data, please contact your teacher.
A sample of the data from a typical file is shown below.
Data Logging with Logger Pro
To complete this activity, you need to access theLogger Pro program.Logger Procan be
accessed from the Toolbox on the online site.
You also need to use the files in the Distance Velocity folder on your course CD. Open this
folder. It contains a series of files including video files and data files.
The video files dist_vel_1.avi, dist_vel_2.avi, and dist_vel_3.avi, all show a video of people
moving either to or from a wall with one person holding a motion sensor and the second person
holding the Lab Quest instrument which records the data taken by the motion sensor and
transfers it to the computer into the Logger Pro program.
The data from the experiment in the dist_vel_1.avi video clip can be found in the dist_vel_1
CMBL file. Similarly the data from the dist_vel_2.avi video clip can be found in the dist_vel_2
CMBL file,and the data from the dist_vel_3.avi video is found in dist_vel_3 CMBL file.
Week 11 Page 10
You can choose to analyse the data from the experiments in either dist_vel_2.avi video clip or
dist_vel_3.avi video clip.
If you open the distance trial folder (and then within this the distance trial MPEG-4 Movie),
you will see Neale Woods, our calculator and Logger Pro guru, explaining how Logger Pro is
used to analyse data. The movie shows data and the position and velocity graphs from this data.
Neale shows how to get the velocity from the position graph.
If you open the dist_vel_1 folder (and then within this the dist_vel_1 HTML document or
dist_vel_1 MPEG-4 Movie), you will see Neale giving a detailed explanation of analysing the
data obtained in the dist_vel_1 video clip. You are asked to do the same sort of analysis of
either the data from either dist_vel_2 or 3 video clips.
The analysis is up to you. You could discuss aspects of the graphs e.g.:
• You might like to describe the path taken by the people in the video.
• How far from the wall were the people at the start of the data taking, and at the end?
• Did they rest?
• What was their speed/velocity over certain sections?
• What was their average velocity over certain sections?
It is suggested that you copy and paste photos, graphs and tables into a Word document to
support your analysis. Include calculations for the displacement and velocity values for the
experiment you have chosen.
In the file dist_vel_1_final CMBL file, you can see Neale’s final analysis of the data from the
dist_vel_1 experiment.
The report (approximately two pages in total length) should include an introduction explaining
the experiment, a main part with the explanation of the models you found and a conclusion
stating what you learned.
Week 11 Page 12
PART 4: Probability
1. If the probability it rains on any one day in July is 0.1, estimate the number of days on
which it will rain next July.
2. The following probabilities relate to two events A and B.
Pr(A) = 0.5, Pr (B) = 0.5 and Pr (A?B) = 0.8
Use this information to determine and hence explain whether or not A and B are:
(a) Mutually exclusive?
(b) Independent?
3. If the probability of a male birth is 0.6, what is the probability that a couple has 3 children
with at least each gender represented?
4. If Pr(A) = 0.25, Pr(B) = 0.35 and Pr(A?B) = 0.4, find:
5. When Sue and Mavis get together they either play scrabble or table tennis. The
probability Sue wins at scrabble is 0.3 and at table tennis is 0.8. The next time they meet
the probability they play scrabble is 0.6.
Draw a tree diagram to show all possible outcomes.
Find the probability that at their next meeting:
(a) Sue wins at whatever they play.
(b) Mavis wins, given that they play table tennis.
6. FOURballs are selected from a bag containing 15 red and 5 blue balls. What is the
probability that:
(a) exactly 3 are red if the balls are replaced each time?
(b) exactly 3 are red if the balls are not replaced?
7. If one card is randomly selected from a standard deck of 52 cards, find the probability of
(a) a red card, given that it is not a heart
(b) a red six, given that it is not a black card
Week 11 Page 13
8. The Venn diagram summarises information on the
sports preferred by a class of 25 year-11 students;
namely, tennis (T), football (F) and basketball (B). If a
student is randomly selected, what is the probability
the student plays:
(a) football, given that the student plays basketball
(b) basketball, given that the student plays tennis
(c) football, given that the student also plays both
the tennis and basketball?
9. If Pr(A) = 0.6, Pr(B?) = 0.7, and Pr(A??B?) = 0.1
Find Pr(A?B)
Are A and B mutually exclusive? Explain.

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