TOPIC:

Functions and Trigonometry

MARKS: Allocated on Assignment

SUBMISSION REQUIREMENTS Term 2 Week 2 Assignments handed out - Date: 10/5/2019 Term 2 week 5 Assignment Due – Date:31/5/2019

WEIGHTING: 30 %

OUTCOMES TO BE ASSESSED: • uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions

to problems MA11-1 • uses the concepts of functions and relations to model, analyse and solve practical problems MA11-2 • uses appropriate technology to investigate, organise, model and interpret information in a range of

contexts MA11-8 • provides reasoning to support conclusions which are appropriate to the context MA11-9

DIRECTIONAL VERBS: Analyse – identify components and the relationship between them, draw out and relate implications Calculate – ascertain/determine from given facts, figures or information Interpret – draw meaning from

TASK DESCRIPTION:

Questions for Assessment • All solutions must be written out in space provided on the question sheet for part 1. • Students will need to print out and hand in the answers to Part 2 and 3. • All solutions must be handed in at the beginning of the class.

ASSESSMENT CRITERIA: • All solutions should be answered on Assignment paper. (part 1) • Answers to part 2 should be emailed or printed and submitted with the assessment booklet. • Answers to part 3 need to be printed and submitted with the assessment booklet.

CAMDEN HIGH SCHOOL Tradition Opportunity Innovation Success

Year 11 - Advanced Mathematics

Assessment Task 2 – Term 2 – 2019

Part one:

Mathematical Applications of Functions and Trigonometry to Crime Scene Investigation

One application of Mathematics is in the field of crime scene investigation. In this part of the assignment you will be asked to solve clues in the form of functions to solve a crime. You will also be applying the methods of trigonometry to evaluate evidence at crime scenes.

a) Using Functions to solve a crime

Detectives, Check your Twitter because the international evil genius terrorist group the Mathemagicians is causing havoc on yet another world region. It seems they are making progress in their plans to build a world conquering device despite our best efforts to apprehend their cronies. West Africa is the victim of the latest string of heists at the hands of a henchman (or henchwoman), Megatransform. One thing is sure, Megatransform is definitely some kind of henchperson.

As has become the calling card of the Mathemagicians, they have left behind a trail of notes with a cryptic text message. The text message when substituted correctly will calculate to Megatranform’s favorite number. So far there are six suspects that police have questioned. It is hoped that someone with a relatively strong number sense and spatial reasoning can crack some codes that have puzzled the detectives on the case so far.

Your job is to bring Megatranform to justice and save the planet. You need to be prepared to state your case and demonstrate your understanding of the following skills that Megatransform is known to use in the notes.

- Families of Functions

- Piecewise Functions

- Modelling – Transformations

- Odd & Even Functions

In your investigation, be sure to show all of your work. We need to have clear evidence that supports your calculations and conclusions. This is not a time to be sloppy. The slightest miscalculation or illegible footnote could result in a not guilty verdict.

Oh, did I mention that use of a calculator might prematurely set off his world conquering device?

Good luck to you.

Scene #1 The Grand Mosque –- Djenne, Mali (5 marks)

Considered one of the greatest architectural achievements, Megatransform apparently took samples of the adobe of the Grand Mosque, which may be used in the World Conquering Device. Later, investigators found this note carved into the floor.

Scene #2 Loango National Park –- Western Gabon (4 marks)

At a relatively new eco-tourist destination, Megatransform crossed over wildlife preserves and chimpanzee-napped an entire family of chimps.

Scene #3 Maison des Esclaves -- Goree Island, Senegal (4 marks)

A security guard reported a break-in this morning at a museum devoted to the history of the Atlantic Slave Trade. Museum curators were startled to find that maps of the Atlantic Ocean were stolen.

Scene #4 Abuja Stadium –- Abuja, Nigeria? (8 marks)

A note was left in the home team locker room of the new state of the art sports complex.

Megatransform apparently has stolen the cleats of the Nigeria National Soccer Team.

Scene #5 Mount Cameroon –- Cameroon (5 marks)

A park ranger believes she witnessed an individual repelling into the volcano (also the highest peak in

West Africa) and left with five bags full of magma.

Scene #6 Ahmed Baba Institute -- Timbuktu, Mali (4 Marks)

Megatransform snuck into the Institute and escaped with important historical documents known as the Timbuktu Manuscripts. Later, investigators were sent a cryptic text message.

(1 Mark)

b) Use TRIGONOMETRY to gain forensic clues and solve cases!

1. A bullet ricocheted off a wall at a 90 degree angle.

Ballistics calculated that the bullet had travelled a total distance of 6m and investigators measured the distance between the two bullet-holes as 3.2m. How far away was the shooter from the final point of impact? (1 mark)

2. A killer left high-heeled shoe prints. The length of the print is 20cm and the shoe style rises at an angle of 12 degrees. How big are the killer’s feet? (1 mark)

3. A stabbing took place with a 20cm blade. The blade penetrated the victim’s chest to a depth of 14cm. At what angle did the blade enter the body? (1 mark)

120 20cm

x

4. A victim was found washed up on a beach, and had clearly been pushed from the nearby pier. The pier makes a 90 degree angle with the beach, and the ocean currents coming towards the beach make a 40 degree angle with the pier. The victim washed up 12m from the pier. How far down the pier should the investigators look to find the place the victim was pushed from? (2 marks)

5. One business set up security cameras to discourage burglars. The cameras are set beside a path, and can record anything that is up to 30m away. The cameras are set to track movement and have a pan of 90 degrees. If a typical burglar walks at about 2m per second, how much footage will each camera capture as they walk along the path? (1 mark)

6. Sarah Strongarm was found strangled. Her elbows were bent 30 degrees as she had tried to brace herself against her attacker and her upper arms are 50cm long. Samuel Shortarm is the main suspect, but he claims that his tiny, 40cm arms could never have reached her neck! Is this true? (2 marks)

7. A door chain allows an 80cm-wide door to open up to an angle of 15 degrees. How long is the chain?

Warning: the door and the chain will not make a 90 degree angle. (3 marks)

d) Bullet Trajectory

An unfortunate reality in our world is that homicides happen. In order to bring justice for the victims, crime scenes have to be carefully dissected to bring about evidence to find and convict a criminal. One way this happens is through a careful analysis of the blood spatter. The shape of a blood spatter tells a great deal about the angle in which it hit the ground. This can enlighten investigators when they are trying to determine the height or distance that the crime took place.

1. Consider these two blood spatters. Which one these do you believe would have hit the ground at a 90˚ angle (meaning it drops straight down)? And which do you believe hit the ground at an angle of around 10˚? ( 1 Mark)

Finding the Angle of Impact

The diagram above shows a hypothetical blood droplet and how forensic scientists piece together evidence using trigonometry.

2. Find the angle of impact of the blood splatter below. Use a ruler to find the measurements of the length and width and use them to calculate the angle of impact. The width is measured at the widest point and the length is the longest point. (2 marks)

Area of Convergence

While the angle of impact is valuable, the area of convergence is often the

more significant piece of evidence. Through seemingly simple forensics and

trigonometry, investigators can estimate the height of the impact and the

distance from the target.

Suppose you walk into a crime scene with these blood spatters:

1. Investigators use fishing line or string to extend each of the blood spatters until they all meet at the area of convergence.

2. Given the angle of impact and this central point, investigators can recreate height with right triangles.

Use the area of convergence and the angles of impact to estimate the height of the incident. Be sure to convert between units as necessary.

1. An investigation is analysing two blood splatters and the information detailed below.

Spatter width Length Angle of impact

Distance from area of

convergence

Height in metres

A 1.9 cm 3 cm 2.7 m

B 1.7 cm 3.6 cm 2.9 m

C 1.8 cm 3.3 cm 2.8 m

a. Calculate the angle of impact for each blood splatter (3 marks)

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b. Use your calculations to estimate the height of the impact. (3 marks)

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c. Considering the height the victim was struck, provide a range of heights that estimate the height of the killer. (1 Mark)

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e) Mobile phone Triangulation

As the world becomes increasingly reliant on mobile technology, we have never been more connected. As a cell phone searches for a signal, its location is pinpointed and coordinated across a number of applications. Your precise location is used on a Google Map or even Facebook. Mobile towers can determine how strong a signal is to a specific phone (i.e. how far away the phone is from the tower) and the general direction the phone is from the tower. The Sine Rule and the Cosine Rule can be used in these cases to get a better sense of where an individual phone may be located.

In each of these problems, information is relayed from two mobile towers

(the western and the eastern), but there is missing information that is needed.

These towers are along the same theoretical horizontal line as seen on the right.

1. Two Directions. The western and eastern cell towers are 10 km apart. From the western cell tower, a

phone signal is N 10˚ E. From the eastern cell tower, the signal is N 20°W. How far away is the mobile phone from each of the towers? Sketch a diagram. (3 Marks) _______________________________________________________________________________________

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2. One Direction, One Distance. The distance between two cell towers is 42 km. From the western cell tower, a phone signal is N70°E and is 48 km away. How far is this mobile phone from the eastern mobile tower? Sketch a diagram. (3 Marks)

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3. Two Distances. The distance between the two towers is 38 km. From the western mobile tower a phone is 21 km away, and from the eastern mobile tower the same phone is 52 km away. Determine the angles and the directions from each of the mobile towers. Sketch a diagram. (3 Marks)

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In the present, important ethical questions about surveillance are being asked. The idea that people can be watched at any time is highly controversial. However, some agree that there are some avenues where this could be a good thing.

The world of forensics is glamorized on TV shows like CSI and often is associated with dissecting a crime scene, combing for hair samples and swabbing for a trace of DNA. In the tech era, mobile devices are used as forensics evidence in court. Since mobile phones can be tracked at all times, proving that a suspect was at a certain location at a certain time has become more scientific. To date, thousands of criminal cases have been proven with forensic mobile phone evidence. Some in law enforcement are optimistic that advances in technology may mean that at some point there could be a foolproof system where a criminal cannot get away with a crime.

In this assignment you must use the Sine Rule and the Cosine Rule to determine where a mobile phone travels based on its records. (Please Note: since the world is a sphere and not a plane, this activity is only a simulation and a more simplified version of what professionals actually do.) (6 Marks)

Part 2: Function Art

1. Go to https://www.desmos.com/calculator and create a free account. This is necessary so that you can save your work. Be sure your username and password is school appropriate, and that you make note of it.

2. Use this link below if you have questions about how to create a particular graph.

a. https://desmos.s3.amazonaws.com/Desmos_User_Guide.pdf HINT: Make sure that you read about sliders, they will help you quite a bit and can be used with either variables in the function or in the domain limits.

3. To limit the domain of a function:

a. Type the function in, and then in curly brackets, type the limitation. For example if I want the quadratic parent function just between -2 and 2, I type: 𝑦𝑦 = 𝑥𝑥2{−2 ≤ 𝑥𝑥 ≤ 2}. Try it and see how it works.

4. To shade a graph write the function as an inequality.

5. Refer to the attached list of Function and Relation Types.

Here’s a fairly simple example of a design at: https://www.desmos.com/calculator/6pwfnvhdei (note the use of sliders and of domain limits)

Plan Your Art 1. Make sure that you READ THE RUBRIC so that you understand the project requirements. 2. What, you say? You are not an artist? OK … let’s brainstorm. Maybe it is a logo for your favorite

school or team. Maybe it is a “shield” that represents who you are. Maybe it is a simple rendition of a piece of famous art. Maybe it is a cartoon character that you want to bring to life.

3. Create a simple pencil sketch of your planned art on graph paper. Here’s the catch … as you create your sketch, think about the functions you have learned (and their transformations). Some of you may prefer to try out ideas directly in DESMOS rather than starting with a sketch….it’s up to you!

Create your Art on Desmos 4. Log in so that you will be able to save your work. 5. Note that you can vary the window if you’d like. 6. You can change the colour of the lines you use. Hold your cursor down on the colour button next

to the equation and a menu of colors will pop up. You can change your line from solid to dotted, and change the colour.

Submit your Work When your artwork is complete, use the share your work icon in the top right corner of the toolbar and email your graph to danielle.brien@det.net.edu.au or print it out, and submit with your assignment.

Function/ Relation Types

For all functions listed below, m, a, b, h, k and r represent constants (real numbers of your choosing)

1. Y = mx + b (linear)

2. Y = a (horizontal)

3. X = a (vertical)

4. Y = x2 (basic parabola)

Vertex form: 𝑦𝑦 = 𝑎𝑎(𝑥𝑥 − ℎ)2 + 𝑘𝑘

Remember : (h,k) is the vertex and a determines width, -a turns it upside down

5. “Sideways” parabola 𝑥𝑥 = 𝑦𝑦2

You can modify the sideways parabola by doing (𝑥𝑥 − 𝑎𝑎) = (𝑦𝑦 − 𝑏𝑏)2

6. Exponential 𝑦𝑦 = 𝑎𝑎(𝑏𝑏)𝑥𝑥 when b > 1, growth, when 0 < b < 1 decay.

a is always positive

7. Circle 𝑥𝑥2 + 𝑦𝑦2 = 𝑟𝑟2 where r is the radius

To shift the circle try: (𝑥𝑥 − ℎ)2 + (𝑦𝑦 − 𝑘𝑘)2 = 𝑟𝑟2

8. Hyperbola 𝑦𝑦 = 𝑎𝑎𝑥𝑥

DO NOT FORGET INEQUALITIES WITH ALL OF THE ABOVE – they’ll give you shading

Try 𝒚𝒚 ≥ 𝒙𝒙𝟐𝟐, 𝒚𝒚 > (𝒙𝒙 − 𝟑𝟑)𝟐𝟐 + 𝟒𝟒 etc.

To colour in a circle (𝒙𝒙 − 𝒉𝒉)𝟐𝟐 + (𝒙𝒙 − 𝒌𝒌)𝟐𝟐 ≤ 𝒓𝒓

RUBRIC Desmos Graphing Project

3 2 1 /0 Score

Uses 5 out of the 8 parent functions

Uses 6 or more than 5 function types

Uses 5 out of 8 functions types.

Uses less than 5 function types

Includes at least two inequalities so that the graph has shaded portions

Uses 3 or more inequalities

Uses at least 2 inequalities

Uses less than 2 inequalities

Includes at least 15 graphs (note that while sliders are extremely useful, they do not count towards the number of graphs)

Uses 20 or more graphs

Uses at least 15 graphs

Uses less than 15 graphs

Creativity and attention to detail

Excellent creativity and attention to detail

Very good creativity and attention to detail

Fair creativity and attention to detail

Total score (out of 12marks): _________________________

Part 3: Just “Tri” Living Without “Trigonometry”

Adapted from the Galileo Academy of Science and Technology

Objective: You will use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.

This project is designed to help you fully grasp the concept of using trigonometry to solve problems. This includes finding the length of a missing side or a missing angle of a right or non-right angle triangle, identifying these problems in everyday life, drawing them, calculating their values, and weaving it all into an interesting, amusing, or creative Question!

Your task is to develop a Question involving the missing angle or missing side of a right triangle. To support your Question, you will need to include a photo that displays the trigonometry used in your story. You will need to include all formulas and all of the steps required to come up with your answer for your Question.

Requirements For this project you should complete two drafts:

• Concept idea (rough draft): o Rough draft of Question, picture, and calculations o Can include color, but not required o Concept idea will help to ensure appropriateness, accurateness, and clarity of thought for

the final draft o This WILL NOT be collected, it is so you can check with me that you are on the right

track. o This draft is for your benefit to insure you have done the project correctly

• Final Draft must include: o Imaginative o Appropriate subject o Must be a trigonometry problem (finding a missing side or angle) o An actual picture to illustrate your Question

The picture must be visible and clearly definable The picture must be appropriate to the question and school expectations.

o A diagram of the right triangle to solve the problem Provides a clear representation of the problem Include realistic measurements Include units Neat (MUST use straight edge, or computer representation) It can be incorporated into the picture but it must be clear legible and visible It must be a right triangle, and labeled with a right angle mark

o Calculations Show formulas you used Show all steps Include units in your answer

o An appealing presentation Professional look Question should be typed or neatly printed Colorful Appropriate subject

RUBRIC “Tri” Living Without “Trigonometry” Project

3 2 1/0 Score

Question Question is of an appropriate length and contains all the relevant information needed to solve the calculation

Question is of an appropriate length, but is missing a piece of relevant information needed to solve the calculation or is unclear.

Question lacks relevant details to solve question.

Picture Is clear with realistic measurements. contains units and all labels

Picture is missing one minor component.

Picture is lacking details or has unrealistic measurements or units

Calculations All working included and calculated correctly matches the story

Minor error in calculations.

Working not shown or doesn’t match story or picture

Presentation Excellent Presentation and attention to detail

Very good Presentation and attention to detail

Fair Presentation and attention to detail

Total score (out of 12 marks): _________________________

Here is an example of the “Tri” Living Without “Trigonometry”Project.