People board the ride at the ground (sinusoidal axis) and the highest and lowest heights you reach on the ride would be the To answer the Ferris wheel problem at the beginning of the section, we need to be able to express our sine and cosine functions at inputs of time. In maths, you have real life applications on any thing that you study. Part 1, Part 2 and Part 3 exams are one hour each, the Part 4 exams are comprehensive and two hours long. Suppose the Ferris wheel begins to decelerate at the rate of 0. Some excellent answers on the $\sin x$ and $\cos x$ functions and how they're solutions to the relevant differential equations were already given, but an important point can still be mentioned: Sine and cosine are used because they are periodic and signals/waves are usually considered to be or are approximated by periodic functions. Search this site. Key Points Unit Circle Trigonometric Functions Exact Values Using Points on the Circle Trigonometric Functions of Angles Exact Values for Quadrantal Angles – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. 5 m above the ground. It completes a full cycle every 120 seconds and starts at the lowest point. III. a) Find a sinusoidal function h(t) that gives the height h, in meters, of the rider above ground as a function of the time t in minutes. To do so, we will utilize composition. 7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Answer any questions with the problem. When the last seat is filled and the Ferris wheel starts, your seat is at the position shown in the figure. and a picture or model of a Ferris wheel. youtube. Work on application problems. Determine the diameter of the Ferris wheel. Along the coast, the tides are of particular interest. Let t be the Martin, Sue. Find the general solution of the equation. asked by Alphonse on October 26, 2012; trig Subsection Modeling with Circular Functions. The wheel rotates twice in 22 min. Find a formula for fC) You have attempted this problem 4 times Your overall recorded soon is 0%. Page 1: A Title page with a short paragraph that summarizes the project. Ferris Wheel Trig Problem tutorial of Trigonometry course by Prof Salman Khan of Khan Academy. ¶ The task also asks students to trace the path of a car on a ferris wheel, precisely, point by point, for a given domain. We model cyclical behavior using the sine and cosine functions. and . The tests are organized by parts. The wheel rotates once every three minutes. Trakimas Math WHS. See more ideas about Med school, Charts and Chemistry help. 13. The highest point on the wheel is 246 feet and initial height is 20 above the ground. Ferris Wheel Function. A Summary of Concepts Needed to be Successful in Mathematics . Opening In the previous chapter, the trigonometric functions were introduced as ratios of . The procedure for graphing sinusoidal functions is similar to quadratic functions. As they come into sight, I realize that some of my circles are simply chosen for decoration and others use properties of circles. 7) Given a pointP (x,y) on the unit circle corresponding to an angle of t, find the sine and cosine. Martin's Classroom 3/28 Sinusoidal Functions Notes and HW Answers. You are riding a Ferris wheel near the Golden Gate Bridge. Materials: One graphing calculator with the capability to graph functions for each small group. F. The starting point of the person is in the lower left hand corner region and it takes 3 second to reach the top, going counter clockwise. 3 Periodic Behaviour - Cycle, period, amplitude p. Nov 20, 2017- Explore rawrstark's board "study 12" on Pinterest. On completion in late 2009, it was to be the highest and largest Ferris wheel in the world. 8 minutes for the wheel to complete one revolution. 0 kg rider at the lowest point of the ride? Teacher guide Ferris Wheel T-1 Ferris Wheel MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to: • Model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions. The key points will indicate the location of maximum and minimum values. You are the last seat filled and the ferris wheel starts immediately. Retrying. The following sheets list the key concepts which are taught in the specified math course. Assume the center of the ferris wheel is on the y-axis, and that the ferris wheel turns 1 revolution every 20 seconds in the clockwise direction. (A) Sketch a graph of the sinusoid. MATH 112 . You probably already know that the MATH 152 COLLEGE ALGEBRA AND TRIGONOMETRY TEST 2 REVIEW TO THE STUDENT: To best prepare for Test 2, do all the problems on separate paper. 1) Ferris Wheel: As you ride a Ferris wheel, your distance from the ground d varies sinusoidally with time t. The wheel starts turning when Percy is at the point P, making an angle of — radians with the vertical, as shown. Man on a Waterwheel Problem Perhaps you have seen the Tom Cruise movie, The Last Samurai. This is how I like to introduce sine and cosine graphs this unit (after spending time with the unit circle and rotations it is a great way to see how we get the sinusoidal graph from a circle, see my blog post here for details ). I want them to think about if the graphs are functions because the path of the Ferris wheel does not pass the vertical line test, so some students may mistakenly say that it is not a function. You are going for a ride on a Ferris wheel in the US (measurements in feet!). 7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. the Perris wheel, your distance the ground Varies. The figure at right shows the locations indicated by \(\theta = 0\degree,~ 90\degree,~ 180\degree,~ \text{and}~ 270\degree\text{. The diameter of the wheel is 40 feet. Explain your answer. Even if students draw the Ferris wheel graph incorrectly, they will still be able to find the period and amplitude. b. Determine the equation of the sinusoidal regression function for the data. Identifi' the graph of your equation from part a. Question: Suppose Our New Ferris Wheel Is To Have A Diameter Of 20 Meters. Give an equation for a transformed sine function with an amplitu The height, h, in metres, above the ground of a car as a Ferris wheel rotates can be modelled by the function. equation yields two solutions, we need additional knowledge of the angle to choose . They are affected by the gravitational pull of both the moon and the sun. With the equation, the height is determined and the times are determined when a person is at a specific height. April . Suppose the diameter of a Ferris Wheel is 42 feet and travels at a rate of 3 revolutions per minute. jpg. FERRIS WHEEL. c) How high above the ground are you after 25 s? Evaluate Trigonometric Functions Evaluate trigonometric functions step-by-step. y = sin x. . As of September 3, 2019, all WORD Problem tutorials have been reprogrammed as lessons with answers . 10, 11 Cart Travel Time p. Round answers to two decimal places if necessary. 359 #1, 2, 3b, 4, 5 2 5. What are the functions of a wheel and axle? A real life example of the sine function could be a ferris wheel. Sandy gets on the ferris wheel at its lowest point and the wheel starts to rotate. a. 21 45 0. Let t be the number of seconds that have elapsed since the ferris wheel started. Find the model that gives your height above the ground at time t (t=0 when you entered). y = cos x. 8. Round values to the nearest tenth. MGSE9-12. When you write A Ferris wheel with a radius of 25 feet is rotating at a rate of 3 revolutions per minute. 71 -0. The radius of the Ferris wheel is 20 feet. 3. Representing the Path of a Single Ferris Wheel. If you’ve ever taken a ferris wheel ride then you know about periodic motion Tides are a periodic rise and fall of water in the ocean. Analyze functions using different representations MGSE9-12. A Ferris wheel with radius 40 feet completes 1 revolution every 60 seconds. If you're seeing this message, it means we're having trouble loading external resources on our website. that the wheel makes revolution once every 8 seconds. Sketch the graph of your height vs. As a Ferris wheel rotates, the vertical height of a seat varies as a sine function. 3 2 S d. 5 m = 53 m. Ferris Wheel problem: As you ride the London Eye ferris wheel, your distance from the ground varies sinusoidally with time. 4d. We are supposed to use sinusoidal functions I believe? The wheel makes 6 complete revolutions every 60 seconds. The direction of rotation is counter clockwise. The URLs for the pages in this new format are not the same as before. We can use angles in standard position to describe your location as you travel around the wheel. Use the graph to answer questions. For this exercise, your class will combine their knowledge of sinusoidal functions as well as linear motion to come up with parametric equations that model the position of a point on the The Task. A ferris wheel is 50 ft in diameter, with the center 60 ft above the ground. 3/15 Ferris Wheel Notes Day 1. On the Ferris wheel problem, students had to imagine that they were riding the . Fe rris W heel Pro blem s 1. For example, we began this chapter with a Ferris wheel of radius 100 feet that rotates once every 8 minutes. As a Ferris wheel turns , the distance a rider is above the ground varies sinusoidally with time. In part (c), most candidates were able to sketch a somewhat accurate representation of the height of the wheel over two full cycles. can see the answers by examining the unit circle, as shown in Figure 5. Since the sine function takes an input of an angle, we will look for a function that takes time as an input and outputs an angle. IFig 2-12 PreCalculus II-Height on Ferris Wheel IFig 2. Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. Solving Trig Equations Using Inverses 15 Helpful Examples. 1) As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. 22 Solutions to Graphing and writing equations of sinusoidal functions worksheet. 3C) Predict the coyote's distance from the ground when the boulder has rolled 1000 feet. Ferris Wheel Problem: As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. m. 2. A vital tool for moving objects around in the model are the isometries, or distance-preserving transformations. Then use the information to find the critical points and sketch two cycles for the graph (one to the right and one to the left of the center point). Understand How The Graph Of A Sinusoidal Function Stretches. (assume you start the ride when you get on). FERRIS WHEEL The lowest point of a Ferris wheel is at the position (O, 0) and the highest point of the Ferris wheel is at (0, 40). The center of the wheel is 24 feet above the ground. 2) + 21, where y represents the height in feet and x represents the time in minutes. Let the origin be the center of the wheel. We start by revisiting the Ferris wheel. 50 Q U I Z 2 H20 : Initial Motion from the Ferris Wheel p. The points on the circle in the diagram to the right represent the position of the cars on the wheel. . One method of graphing sinusoidal functions is to find five key points. c. 2C) Write a cosine function expressing distance from the ground as a function of distance travelled down the mountainside. This is the fourth section, and it applies many concepts from the previous sections (especially Sections 1 and 2) to periodic functions that appear in real life. Label the A real life example of the sine function could be a ferris wheel. Can you sketch an angle in standard position given its radian measure? 1. To answer the Ferris wheel problem at the beginning of the section, we need The Khan Academy has video material that walks through this problem, which you may find easier to follow: Ferris Wheel Trig Problem · Ferris Wheel Trig 24 May 2017 Donna is riding a 100 foot diameter Ferris Wheel with a center Are both of my answers here (the equations and the coordinates at the Now, since sine is a odd function, this means \sin(-x)=-\sin(x), and so we now have: You can graph sinusoidal functions using your knowledge of transformations. For example, an application might involve sin t, where t represents time. From this information, you can find the value of constant b. Graph the height functions you found in #3 on the same graph using the time interval: The general forms of a sinusoidal equation are given as Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona. The wheel rotates at a constant rate, taking 1. 6) sin t=y (5. Moving Cart, Turning Ferris Wheel p. Students were then asked to predict what would change about the graph and the equation if a platform had the people enter at 3. 1536 p. a) What is the diameter of the wheel? Download Free Periodic Trig Function Word Problems and Answers . Items 1 - 7 functions as well as trigonometric identities and reciprocal identities. pdf. You find that it takes you 3 seconds to reach the top, 43 ft above the ground, and that the wheel makes a revolution once every 8 seconds. 19 35 1. Sinusoidal regression Unit 8 Sinusoidal Functions Students went to the verticals to work on a problem involving a ferris wheel. Write an equation in rectangular form of the graph of the Ferris wheel. your functions, but otherwise round answers to 3 decimal place accuracy. the radius of the Ferris wheel The height, h, in feet of the tip of the minute hand of a wall clock as a function of time, t, in minutes can be modeled by the equation mc017-1. The Period goes from one peak to the next (or from any point to the next matching point): The Amplitude is the height from the center line to the peak (or to the trough). They were to find the equation of the height of a car on the wheel. 21 0. Assume that a rider Enters a car that is located 30degrees around the rim before the car reaches it's lowest point. Interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time. a) Determine the equation of the sinusoidal function, h(x), that models the height of a passenger on the ride as a function of time. A GPS receiver measures distance using the travel time of radio signals from satellites. Welcome to Mrs. SINUSOIDAL APPLICATION PROBLEMS from Paul Foerster. This course offers over twenty lectures that include word problems to calculate functions of angles, and other simple applications of trigonometry such as pendulum, wind turbine, helicopter and ferris wheel word problems. b) Find the height h after 45 seconds. It This is not your typical Ferris wheel problem. (B) What is the lowest you can go as the Ferris wheel turns, and why is this number greater than zero? (C) Write an equation of this sinusoidal. 1. Texas Instruments, regardless of the form of action, shall not exceed the purchase price of this product. Write a model for the height h (in feet) of the chair as a function of the time t(in seconds). When will you be high enough to see the full view? Students attempt to answer this question using some knowledge of right triangles Plan your 60-minute lesson in Math or Trigonometric functions with helpful tips from Hilary Yamtich Inverse Trig Functions: Arcsin Unit Circle Definition of Trig Functions Trigonometry problems dealing with the height of two people on a ferris wheen Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Let t be the number of seconds that have elapsed since the Ferris wheel started. Show your work. To define our trigonometric functions, we begin by drawing a unit circle, a circle . Sydney wants to ride a Ferris wheel that has a radius of 60 feet and is suspended 10 feet above the ground. Assume the person gets to ride for two revolutions. In Canada’s wonderland there is a roller coaster that is a continuous series of identical hills that are 18m high from the ground. It requires that you state the transformations, determine the mapping rule, transform the table of values, find the equation and graph the results. 5. IF. (Figure) lists some of the values for the sine function on a unit circle. Sinusoidal Ferris Wheel Problem? I've done this problem before, but I'm a little confused as to the method. If the ride makes three complete rotations, the total amount of time a rider on the Ferris Wheel will spend above 13 m, rounded to the nearest second, is (A) 11 s (B) 15 s (C) 25 s (D) 32 s (E) 45 s IV. D. This common word problem always seems tricky, but we show you how to break the question down to develop a trig equation. Write an equation about the movement of a Ferris wheel. (That will explain any typos). Question 1 : The angle of elevation of the top of the building at a distance of 50 m from its foot on a horizontal plane is found to be 60 degree. As you are waiting in line to ride the Ferris wheel at the state fair, you begin to be hypnotized by the repetitive motion of the seats. Kumon answers, "power point templates" freeware, mathematic calculator for tax. Finally, give your students exercises involving distances and times through which Big idea In this unit you will learn how trigonometry can be used to model wavelike relationships. Write a cosine and sine equation for the graph. Write the trigonometric equation for the function with a period of 6. The graph is shown as a Sine curve and function or a Cosine curve and function. }\) There are many uses of sin,cos,tan in real life. There is nothing we hate more than a chapter on exponential equations that begins "Exponential functions are functions that have the form f(x)=ax. 1 - Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: World’s Largest Ferris Wheel Example on pg. b) The sinusoidal axis is 14m. Section 4 — Outline Lesson 1 Modelling with Trigonometric Sinusoidal functions are often useful in modeling data that has been collected and plotted. Now, the height is the y axis. WHY THESE SHEETS ARE USEFUL – Imagine that you are riding on a Ferris wheel of radius 100 feet, and each rotation takes eight minutes. Periodicity. 2 Use phase shifts of sine and cosine curves. MODELING PERIODIC PHENOMENA WORKSHEET Behaviors that can be modeled by a sine or cosine function – moon phases, ocean tides, amount of daylight, Ferris wheel (or anything that moves in circles) sinusoidal function – a function that can be represented in the form y = A sin [b(x - h)] + k or y = A cos [b(x - h)] + k 8. The six o'clock position on the Ferris wheel is level with the loading platform. At the highest point, a seat on the Ferris wheel is 46 feet above the ground. Problem Set 6. In one scene, a man is tied to a water wheel. Since most problems state the position of a maximum or minimum, it is probably wise to use COSINE functions rather than SINE functions. Write parametric equations to model Donna's motion at any time if she is at the bottom of the wheel at time t=0. 68 1. The mathematical content is the topic of sine and cosine functions, and specifically . You are the last seat filled and the Ferris wheel starts immediately. The Bottom Wheel Is 2 Meter Above The Ground. the directions below and answer each question as you complete the activity. " As each family of functions is introduced, we motivate the topic by looking at how the function arises from The Ferris Wheel There are many rides at the amusement park whose movement can be described using trigonometric functions. As such, sinusoidal functions can be used to describe any phenomenon that displays a wave or wave-like pattern or by extension any predictable periodic behavior. When the last seat is filled and the ride begins, your seat is at the position shown. 0 m. Model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions. a) Find the average angular speed of the Ferris wheel in radians per second. Identify the choice that best completes the statement or answers the question. Let t be the number of seconds that have elapsed since the A Ferris wheel has a radius of 18 meters and a center C which is 20m above the ground. When the last seat is filled and the Ferris wheel starts, your seat is at the position shown in figure 2-12d. Okay let me clarify my answers to the question from part a) to e) a) Here is a sketch of the graph. Assume that high tide occurs at 12 noon and at midnight, and that low tide occurs at 6 pm and 6 am. Sinusoidal components appear in functions that describe AC circuits, ocean tides A Ferris wheel 50ft in diameter makes one revolution every 40sec. Students should be able to graph the equation h(t) = -20cos[([[pi]]/ 5)t] + 20 One of the most useful applications of the trigonometric ratios allows us to find distances or . If you're behind a web filter, please make sure that the domains *. At low tide the water reaches the 1-foot mark. If you have three members, then you have to do the extra credit model. The table of values for the functions . Ferris Wheel Ferris wheel project connects trig functions, circular motion, parametric equations, and the distance formula in 2 and 3 dimensions; requires a graphing utility to solve a problem to which everyone can relate. ( amplitude). H14 : Putting the Cart Before the Ferris Wheel p. to 12:15 p. You find it takes 8 seconds to go around one time. A. a) Draw the graph of the situation, starting with a person getting on at the bottom of the wheel at time t = 0 seconds. The radius would be the radius that the seat moves, which is 2 feet less than the radius of the ferris wheel itself. When his stopwatch read 5 seconds, the point was at its highest, 16 feet above the water's surface. Ruby has a pulse rate of 73 beats per minute and a Solve word problems that involve real-world contexts that are modeled by sinusoidal functions. (I re-did the graph) Math 2204/05 Name:_____ Sinusoidal Word Problems Chapter 3 1. Here is the problem: As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. It carries each car around a circle of diameter 19. A ferris wheel with a radius of 25 meters makes one rotation every 36 seconds. The diameter of a large Ferris wheel is 48 meters and it takes 2. c) Find the sinusoidal equation that models your ride on the Ferris wheel. He notices that he is 50 feet above the ground at his highest point. Sketch a graph that represents the given population model. Let t be the # of seconds that have elapsed since the Ferris wheel started. ) Answer Formative Assessment Lesson: Ferris Wheel. As I sit in my family room typing this, I am looking around for circles. The Ferris wheel starts going counterclockwise, and you find that it takes you 5 seconds to reach the top and the wheel makes a revolution every 35 seconds. 1|Graphs of the Sine and Cosine Functions Learning Objectives In this section, you will: 6. If the Ferris Wheel rotates counter-clockwise, instead of the original clockwise motion, the new To answer the Ferris wheel problem at the beginning of the section, we need to be able to express our sine and cosine functions at inputs of time. Home About Calculus 12 and are called Periodic Functions. The company building the Ferris Wheel has decided the Ferris Wheel may run too fast and decreases the rotation speed to 40 minutes. 5 metres and rotates every 20 seconds. Lesson Wrap-up Ferris Wheel problem: http://www. docx from MATH physics problem I need help with this Algebra 2 Answer to Ferris Wheel In Problem Set 2. Draw A Sketch Of A Ferris Wheel, Label All Information 2. Graph data, and determine the sinusoidal function that best approximates the data. A ferris wheel is 50 feet in diameter, with the center 60 feet above the ground. Key Terms correctly in your class discussions and in your problem solutions. 30. Ixl Graph Cosine Functions Year 11 Maths Practice. It rotates once every 40 seconds. a) Write a new equation giving height of a person using the sine function. Find the equation of for d(t). • Develop and use the Graphs of sine and cosine functions are called sinusoids. 11 6 S c. Specifically, they find distances between points of a circle and a given line to represent the height above the ground of a passenger car on a Ferris wheel as it is Problem Practice Polar and Rectangular Forms of Equations 1. Lesson 12: Ferris Wheels—Using Trigonometric Functions to Model quietly about the problem, and then ask for volunteers to answer the question. 244to 247 in Text The “London Eye” is the world’s largest ferris wheel which measures 450 feet in diameter, and carries up to 800 passengers in 32 capsules. 39 H11 : Where Does He Land p. Lesson 3: Radian and degree | Unit circle definition of trig functions; Lesson 4: Using Trig Functions Ferris Wheel Trig Problem; Lesson 24: Ferris Wheel Trig Mathematics 5 SN SINUSOIDAL GRAPHS AND WORD PROBLEMS The tuning fork is a device used to verify the standard pitch of musical instruments. org and *. com - id: 489fd1-OThhO Nova Scotia Mathematics Activities & Resources. In the problem, they give us the information that we need. At the bottom of the ride, the passenger is 1 meter above the ground. The wheel has a meter diameter, and turns at three revolutions per minute, with its lowest point one meter above the ground. ) (in degrees – teacher will provide Jackson gets on a ferris wheel at its lowest point which is 10 feet above the ground. Before beginning this activity, students should have been introduced to sine and cosine. Trigonometry problems dealing with the height of two people on a ferris wheen. 4 Sinusoidal Regression Project In this task, we gather and examine a periodic data set that can be modelled well with a sinusoidal function. The wheel makes one full rotation every 7 minutes, and at time t=0 you are at the 3 o'clock position and ascen Log On The Beijing Wheel The Beijing Great Wheel is an observation wheel currently undergoing construction** in Beijing, People's Republic of China. Investigating Functions with a Ferris Wheel: Distance vs. Section 4 — Outline Lesson 1 Modelling with Trigonometric Problem Solving Introduction Circular functions is a very lengthy topic. Find the equation that gives you your height when you entered the ferris wheel above the ground at t time. Using the axes below, sketch a graph to show how the height of a passenger will vary with time. co. C. Modeling Real World Data With Trig Functions Ferris Wheel Review for Unit 8 Sinusoidal Functions Name:_____ Multiple Choice Identify the choice that best completes the statement or answers the question. The wheel rotates at a constant rate in the direction shown by the arrow, taking 1. A ferris wheel has a diameter of 14. A Ferris wheel can be modeled, in meters, by the equation, r = 12 sin θ. 4C) What are the first four times the coyote is a height of 2 feet from the ground? Triangulation a process that works by using the distance from two known points, is used in cell phones equipped with GPS. The second set of problems deal with tangent curves Loading Loading Graphing Trig Functions Date_____ Period____ Using degrees, find the amplitude and period of each function. The centre of the wheel is 26 m above the ground. When the last seat is filled and the Ferris wheel starts, your seat is at the position shown in the figure above. Assume the wheel has a diameter of 10 m and the center of the wheel is 3 m above the water. You may tear it off to do your test. I visited the park to take a video of the Ferris wheel in action. are transformed to model sinusoidal functions. Martin, Sue. ( Lesson 12 Aug 2014 What is the domain of your Ferris wheel height function? passenger car at the outset of the problem (that is, after a 0° rotation) to be 0 feet. time for one full cycle. Graph A Rider's Height Above The Above The Ground Verses The Time During One Rotation (ride Start At The Top Of The Wheel) 3. One of the largest ferris wheel ever built is in the british airways london eye which was completed in 2000. Hello, all. Suppose Tommy starts the ride in the 3 o'clock position. a) Sketch a graph b) Find the equation for the riders height above the ground. 39 40 1. Functions, Section 5 Worksheet - Applications and Models. Assume that the wheel starts rotating when the passenger is at the bottom. Sinusoidal Functions as Mathematical Models You are riding a Ferris Wheel that is 40 feet in diameter. The radius is 12m. The Phase wheel of a paddleboat is a function of the time, t, in seconds as outlined in the data chart below. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. 32 Ferris Wheel Packet sinusoidal function. At its lowest point, the Ferris wheel is 5 feet off the ground. 6. Exploring Sinusoidal Functions This task serves as an introduction to the family of sinusoidal functions. Given a sine function graph the function using technology. com - id: 59b270-YWRlZ Example 1: In an amusement park, there is a small Ferris wheel, called a kiddie wheel for toddlers. Objective: To determine and investigate different functions that represent the height of a person on a single Ferris wheel. 3m makes 2 rotations every minute. Therefore, it has been split into four sections. The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). 40 the person on the Ferris Wheel? 10. Tide Problem. The diameter is 135 m and passengers get on at the bottom 4 m above the ground. 22. Draw an angle in standard position whose measure is : a. You enter from a platform at the 3 o'clock position. Find the length of time between successive periods of maximum population. g. 5 metres above the ground, rather than at the bottom of the wheel. The high tides and low tides follow a periodic pattern that you can model Equations/1140263: Delilah wants to join a gym, so she shops around to find the one with the lowest overall price (she is not sure how long she will be a member). However, it seems that many candidates are not familiar with the shape of a sinusoidal wave, as many of the candidates' graphs were constructed of line segments, rather than a curve. a) Let h be the height, above ground, of a passenger. It says that the ferris wheel is already off the ground 3 feet when you get on, so we know that none of our points will go lower than 3. The graph of a sinusoidal function Sine or the sin function is one of the three primary functions in trigonometry, the others being cosine, and tan functions. 8 Applications of Sinusoidal Functions. every 3 minutes. So here, I found my two equations, please check to see if they are correct! Ferris wheel sinusoidal function problem? The ferris wheel is 53 feet tall and has a radius of 25 feet. I know this much: I can determine the radius and central angle of the wheel, and I can draw another radius line to make a right triangle with. Problem 6. A "horn" means that the lesson has audio recordings; while lessons without any audio files have the icon . Or we can measure the height from highest to lowest points and divide that by 2. You find that it takes you 3s to A seat’s position on a Ferris wheel can be modelled by the function y = 18 cos 2. Please update your personal links. Sketch two cycles of the described sinusoidal graph. Determine a sinusodial equation that gives her height, h, above . With the equation, the further their knowledge of trigonometric functions and enable them to understand that . The bottom of the Ferris wheel is 2 m off the ground. I encountered a few problems for a few questions while doing my homework. A rider gets onto the wheel at its lowest point which is 60 cm above ground at t = 0. Have the students share their graphs and you can take the time expounding on the various aspects of the sinusoidal graphs the students created (e. A Ferris wheel (a vertical circle for the purposes of this question) with a radius 42. Eureka Math: Module 2 Lesson 4, 6, 7 Math 175: Chapter 6 Review: Trigonometric Functions In order to prepare for a test on Chapter 6, you need to understand and be able to work problems involving the following topics. The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA II (Common Core) Wednesday, June 1, 2016 — 9:15 a. The lowest point of the wheel is 5 feet above ground. asked by CL on March 29, 2010; Physics Problem Solving Introduction Circular functions is a very lengthy topic. If the center of the wheel is 30 ft above the ground, how long after reaching the low point is a rider 50 ft above the ground? Our teacher said to model the situation with an equation. 1 Graph variations ofy=sin( x )andy=cos( x ). Sinusoidal Functions as Mathematical Models WS #1 NAME: 1) Ferris Wheel Problem. McGraw-Hill Ryerson, 2008. 12 allows use of the Height When Riding a Ferris Wheel to construct the graph showing the height above the ground of a point on the Ferris Wheel as the wheel rotates. The platform to get on the ride is on top of the first hill. Answers: 1. 1. The pages referenced are in the textbook and the answers to odd-numbered problems are given in the back of the book. \begin{equation*} \begin{aligned}[t] y \amp = r \sin 205\degree \\ \ amp In order to graph the Ferris wheel function, we must first specify the input and of the graph, and the location of the maximum and minimum values. Transformations Of Trig Functions She Loves Math. SOLUTION Even if students draw the Ferris wheel graph incorrectly, they will still be able to find the period and amplitude. The sine x or sine theta can be defined from a triangle as the ratio of the opposite side to that of the hypotenuse. Give your answer as a range of dates, to the nearest day. You will study key properties that these functions have and use these properties to sketch functions to model real life situations and to solve HS Algebraic Functions A Semester 2 Module 2: Trigonometric Functions Topic A: The Story of Trigonometry and Its Contexts Topic A starts by asking students to graph the height of a Ferris wheel as a function of time and uses that study to help define the sine, cosine, and tangent functions as functions Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. Applications/Problem Solving Using Trig Functions (Ferris Wheel, Tides, Climate, etc. Let t be the number of minutes elapsed since the ride began. 1) A ferris wheel is 4 feet off the ground. Write a sinusoidal function that describes the function from the person’s position. At high tide the ocean generally reaches the 5-foot mark on a retaining wall. Then graph. These wavelike functions are called sinusoidals (they come from either sine or cosine equations). Introduction PRACTICE Trig Word Problems 1. Finding Equations and Graphing Sinusoidal Functions. The rotation of a Ferris wheel. Angular measure problem: A Ferris wheel with a radius of 25. Need help with this problem Thanks The figure below shows a Ferris wheel that rotates three times each minute. As you ride a Ferris wheel, the height that you are above the ground varies periodically. Page 3: Table Of Contents Power, Polynomial, and Rational Functions Graphs, real zeros, and end behavior Dividing polynomial functions The Remainder Theorem and bounds of real zeros Writing polynomial functions and conjugate roots Complex zeros & Fundamental Theorem of Algebra Graphs of rational functions Rational equations Polynomial inequalities Rational inequalities Ferris wheel problem Characteristics of sine and cosine graphs Matching graphs and functions The Graphs of Sinusoidal Functions. Start concretely. 49. Letf) denote your height On meters) above ground at t minutes. Ferris Wheel Trigonometry Problem This video explains how to determine the equation that models the height of person on a Ferris wheel. 6) Activities to Support Instruction: Mnemonic Device for Values of Trig Functions Modeling and Analyzing with Ferris Wheel Problem Modeling and Analyzing with Donali National Park Daylight Hours problem Page 1 of 2 840 Chapter 14 Trigonometric Graphs, Identities, and Equations Translations and Reflections of Trigonometric Graphs GRAPHING SINE AND COSINE FUNCTIONS In previous chapters you learned that the graph of y =a • ƒ(x ºh)+k is related to Sine and Cosine Functions If t is a real number and a point (x, y) on the unit circle corresponds to an angle of t, then cos t=x (5. Consider the height of the center of the wheel to be the equilibrium point. 5 m + 50. Sketch trigonometric graphs for each of the following situations: Situation A A Ferris wheel with a diameter of 6m makes one complete revolution every 10 seconds. 42. They are applicable in many real life cases. R-1. Solution . Students conjecture how these functions are related to the trigonometric ratios they studied in geometry, making plausible arguments by modeling the Ferris wheel with a circle in the coordinate plane. kastatic. People board the ride at the ground (sinusoidal axis) and the highest and lowest What Are Some Examples Using Sinusoidal Functions in Real Life? Credit: Dave Fayram/CC-BY-SA 2. Worksheet 6 6a. Lesson 12: Ferris Wheels—Using Trigonometric Functions to Model Cyclical Behavior Student Outcomes Students review how changing the parameters 𝐴, 𝜔, ℎ, and 𝑘 in ( )=𝐴sin(𝜔( −ℎ))+𝑘 affects the graph of a sinusoidal function. What is the amplitude of the sinusoidal function y = 7 sin -8 X-*. It rotates once every 32 seconds in the direction shown in the diagram. 5 minutes for each full revolution. a) What is the centripetal acceleration of a rider? Answer in m/s^2 (b) What force does the seat exert on a 40. 5 Unit 6 – Trigonometry Higher Level Questions for More Complex Concepts OR an EXTENSION of basic concepts involved with triangle trigonometry and sinusoidal functions. Activity Solve this problem in small groups. You enter from a platform at the 3oclock position. Read the TexPoint manual before you delete this box. A carnival Ferris wheel with a radius of 7 m makes one complete revolution every 16 seconds. Write parametric equations describing the path that Tommy takes on the Ferris wheel. Determine h as a function of time if h = 51 meter at t = 0. The Ferris Wheel is a good example of periodic movement. U B tMBandIeP HwXidtahA QIUnHfGijnHivtjeT rPqrfetcraXlBcuuGlLu]si. Test-outs are three hour exams each. When t= 0, a chair starts at the lowest point on the wheel, which is 5 feet above the ground. Exam Question [] "Jacob and Emily ride a Ferris wheel at a carnival in Vienna. The kiddie wheel has four cars, makes one revolution every minute, and has a diameter of 20 feet. Graph two periods of the function d(t). 4 Graphs of Sinusoidal Functions Handout The path of the seat on the ferris wheel is circular. O L MAUlple erziWg`hgtFsQ Gr`exsye[rOvRerdO. It uses a desmos applet to let students explore the effect of changing the parameters in y=Asin(B(x−h))+k on the graph of the function. a) Draw a graph which represents the height of a passenger in metres as a function of time in minutes. It takes 80 seconds for the ferris wheel to make one revolution clockwise. The next thing that it tells us is that the diameter of the ferris wheel is 38 feet. They analyze givens, constraints, relationships, and goals. TLDR: Here’s the 101questions page. Ferris Wheel A Ferris wheel is 60 meters in diameter and rotates once every four minutes. The following activity is a one day activity dealing with trigonometric functions. Ch. above ground. Write the trigonometric equation for the function with a period of 5, a low point of – 3 at x=1 and an amplitude of 7. 08. How high above the ground would a person be 16 Before starting these examples you might want to refresh you memory on solving these problems. Answer to A little more sinusoid practice! 40ft 20ft Ferris wheel #1 Ferris wheel #2 Problem 2. 4. What I Did. Problem 4: Ferris Wheel Problem As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. The constant is pi/? Please show me HOW to do this problem as well if you can!! I appreciate any Real World Examples of Periodic Functions: Sound waves, like any other waveform, consist of wavelength, frequency, velocity, and amplitude. 5) Phase Shift; Sinusoidal Curve Fitting (5. The wheel rotates at a rate of 2 revolutions every 6 minutes. Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. I. Suppose a Ferris wheel with a radius of 20 feet makes a complete revolution in 10 seconds. Questions are More Important Than Answers. , only SCORING KEY AND RATING GUIDE Mechanics of Rating The following procedures are to be followed for scoring student answer papers for the Regents Examination in Algebra II (Common Core). Sine, Cosine, Tangent Real World Applications. If a rider starts at the lowest point, find the height of the rider after 20 seconds. Unit Circle Definition of Trig Functions - Graph of the sine Ferris Wheel Trigonometry Problem This video explains how to determine the equation that models the height of person on a Ferris wheel. 01 0. Solve, using technology, a contextual problem that involves data that is best represented by graphs of sinusoidal functions, and explain the reasoning PreCalculus Sinusoidal Model Project 2008 READ EVERYTHING FIRST! Group project: two or three members in a group. Graphing Sine and Cosine Functions. a). - 268767 6. She finds the Harbor Square Athletic Club is running a special, and only charges a $25 initiation fee plus $87 a month to be a member. trigonometric function that can be used to answer questions about the situation. We must pay attention to the sign in the equation for the general form of a sinusoidal function. I am surprised that you know them and ask such a question. In many applications, the variable or function argument represents something other than angles. it takes jackson 40 seconds to make one revolution-what is the period of the function-what is the amplitude of the function-what is the midline of the function? MCR3U Grade 11 Functions math Trigonometric Functions Test Math trigonometric functions unit test notes Amplitude: distance from axis of symmetry to the maximum or minimum point. 1 Answer to A ferris wheel 40 feet in diameter makes one revolution every 30 seconds. B. Identify characteristics from a function Getting the function from a sine graph Getting the function from a cosine graph The Equations of Sinusoidal Functions. Now that we have used radians to define the trigonometric functions, we can describe periodic phenomena as functions of time (or other variables besides angles). 3 4 Sinusoidal Functions: Problem 2 point) A ferris wheel is 35 meters in diameter and are at the 9 o'clock position and descending. An easy way to describe these functions is as follows. ISBN: 0070126593 Rates of Change Prerequisite Skills Rates of Change and the Slope of a Curve Rates of Change Using Equations Limits Limits and Continuity Introduction to Derivatives Extension: Use a Computer Algebra System to Determine Derivatives Review Practice Sample problem: Transform the graph of f(x) = sinx to sketch the graphs of g(x) = –2sinx and h(x) = sin(x – 180°), and state the domain and range of each function. You may write on your test, but all answers need to be marked on this sheet. This question is from a Holt textbook. You find that it takes you 8 minutes to reach the top, 443 ft. The wheel's diameter is 18 ft and it completed a revolution every 10 seconds. A Ferris wheel with a radius of 25 feet is rotating at a rate of 3 revolutions per minute. The function has a maximum of 3 at x = 2 and a low point of –1. The wheel starts turning when Percy is at the point P, making an angle of radians with the vertical, as shown. REVIEW SHEETS . Graphs of Trigonometric Functions. kasandbox. 0542 rad/s2. At a seaport, the water has a minimum depth of 4m at 3:00 am. 12) A Ferris wheel has diameter of 60 feet, it’s center is 35 feet off the ground, and it takes 25 seconds to complete one revolution. It moves so slowly that there is usually no need to stop the wheel to let people on or off. The bottom of the wheel is 1. After you finish the multiple choice portion, be sure to complete the short answer questions on the reverse side of this sheet!!! Graphing trigonometric functions lesson plans and worksheets from thousands of teacher-reviewed resources to help you inspire students learning. The centre axle of the Ferris wheel is 40 meters from the ground. Computer lab: check to see if you have access to the HS Drop folder, if not let me know on the first class! At Oregon Fossils - Upsky. Moreover, Texas Instruments shall not be liable for any claim of any kind whatsoever against the use of these materials by any other party. If the center of the wheel is 30ft above the ground, how long after reaching the low point is a rider 50ft above the ground? Whoops! There was a problem previewing HPC - Graphs of Trig. Which number (from 1 to 12) is the minute hand pointing to at t = 0? Trigonometric Equation Calculator (Evaluating Trig Functions ) If you’ve ever taken a ferris wheel ride then you know about periodic motion, you go up and down 6. Below is a 30 second clip of the "Big Ellie" Ferris Wheel at Atlantic Playland. TRIGONOMETRY . 1 C) Sketch a graph of this sinusoidal function. As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. Andrew stands 90 feet to the right of the base of the Ferris wheel and Just like the above, it is an application problem. A boy on a Ferris wheel that turns at a constant rate of 1 revolution every 3 minutes is at most 23 metres above the ground and at least 2 metres above the ground. b) Predict the position of a person on this Ferris Wheel after 8 minutes. pdf 08. Hes been out of college for over 10 years so he was very rusty with his math skills. 6 Applications of Sine and Cosine Functions Worksheet #1 MCR3U Jensen 1) At a maximum height of 135 m, the Millennium Wheel, in London, England, is the largest cantilevered structure in the world. The lowest point on the Ferries wheel is 2. The periodic rotations of a crankshaft in an engine. SINUSOIDAL APPLICATION PROBLEMS from Paul Foerster FERRIS WHEEL 1) As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. 3 4 S b. (16 pts) Percy is riding on a ferris wheel of radius 50 feet, whose center C is 52 feet above ground. A ferris wheel has radius of 25 m and its centre is 27 m above the ground. Saying “it takes 40 seconds to complete one revolution” isn’t the same as seeing a ferris wheel travel at that speed. 2 m is initially rotating at a constant rate, completing one revolution every 33. 66 ACT Practice . sketch a graph. 10. Trigonometry Functions and Graphs Transformations of Sinusodal Functions Graph and transform sinusoidal functions Identify the domain, range, phase shift, period, amplitude, and vertical displacement of sinusoidal functions Recall the following equation from transformations: What the letters a, b, c, and d mean in trig Yrelch Jhí( (R 3(dC) Whoops! There was a problem previewing MCR3U Exam Review 2016. uk Download and Read Oregon Fossils for young children corporate finance final exam answers software to check internet speed pdf ferris wheel problem sinusoidal functions answer key mercruiser engine wiring harness diagram prentice hall study guide answers earth A Ferris wheel has a radius of 25 m. You find that it Sinusoidal Functions as Mathematical Models. org are unblocked. The wheel was designed to measure approximately 500 feet in diameter and to carry up to 800 passengers in 32 capsules. When you first notice the man on Lesson 1: Ferris Wheels—Tracking the Height of a Passenger Car Student Outcomes Students apply geometric concepts in modeling situations. multiple Ferris wheel problems, I am hopeful that these scenarios will . It Will Make Rotation In 20 Seconds. com/watch?v= o7Ho1bMWhG8. 5 m and the highest point is 2. With this dissertation, I answer each of my three main research questions How do you sketch the graphs of sine and cosine functions? What's the Discuss Activity 2 with students using the answers and questions provided. 5 metres above the ground. ) When Paul throws a ball, he ALWAYS releases it at a height of 5 feet and at an angle of 75 degrees. (D) Predict your height above the ground when 𝑡=6, 𝑡=9, and 𝑡=0 The movement of planets around the sun, the motion of a yo-yo are all examples of periodic functions. sinusoidally with time. Graph sinusoidal functions including transformations Determine the equation of a sinusoidal function Solve Trigonometric Equations Solve real-world problems involving sinusoidal functions Day text Topic Practice Doneü 1 5. Suppose a Ferris wheel with a radius of 20 feet makes a complete revolution in 10 a function of time, thereby establishing that the height is a sinusoidal function of t. Academic Write your answers on notebook paper. 5/5. This is your answer sheet. words into meaningful and measurable functions that model the world around us. • The diameter of . Find the sinusoidal function that models this Ferris wheel in terms of time Algebra -> Customizable Word Problem Solvers -> Misc-> SOLUTION: A ferris wheel is 40 meters in diameter and boarded at ground level. You Modeling with sinusoidal functions: Word Problems 1. Ferris Wheel Problem: Picture yourself on a Ferris wheel similar to the picture. These numbers vary based on the air temperature; however, at sea level (68 degrees,) the speed of sound is 767 miles per hour. A Ferris wheel has a radius of 20 m. 0 In the real world, sinusoidal functions can be used to describe mechanical functions such as the swinging of a pendulum or natural phenomena such as hours of daylight. 10 Graphs of Sinusoidal Functions. Give the equations for the Ferris wheel. my answer: y=-25sin(pi/40x)+35 3. Lett be the number of seconds that have elapsed since the Ferris wheel You find that it takes you 3 The Ferris wheel makes one rotation every 24 seconds, with a person sitting 26 metres from the ground and rising when it starts to rotate. If the ride begins at point P, when the time t = 0 seconds: Activity Dealing with Trigonometry Functions . For homework, we got a problem that reads as follows: A Ferris wheel 50 ft in diameter makes one revolution every 40 sec. So what do they look like on a graph on a coordinate plane? Periodic Functions by: Doris Santarone To celebrate the new millennium, British Airways announced in 1996 its plans to fund construction of the world’s largest Ferris wheel. Though the example of a pendulum is a special case of periodic function because it is executing simple harmonic motion, the difference lies in how the motion is expressed mathematically, if the periodic function can be represented by a sine curve then the motion is said to be simple harmonic The vertical and horizontal displacements of a Ferris wheel passenger car are both periodic. Teacher guide Representing Trigonometric Functions T-1 Representing Trigonometric Functions MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to: • Model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions. When the last seat is filled and the Ferris wheel starts, your seat is at the position shown below in the figure. Graphs Of Trigonometric Functions Trigonometry Math. For more trigonometry word problems, sign up for the Trigonometry: Trigonometric Functions II course. 8 Sinusoidal Functions Unit Assignment A seat’s position on a Ferris wheel can be modelled by the function Problem 1. It takes 80 sec to complete one revolution. Construct a graph that models the boy’s height above ground with respect to time. The sheets present concepts in the order they are taught and give examples of their use. Video: Model how a trigonometric function describes the relationship of a Ferris wheel rider as the wheel the given measure and the period, phase, offset and amplitude of a cosine function. 1) y = sin 3 θ 60 ° 120 ° 180 ° 240 ° 300 My husband has been using the software since he went back to school a few months ago. 2 seconds. Passengers get on halfway up on the right side. These points will correspond to intervals of equal length representing 1 4 of the period. Unit 5: Graphing Trig Functions & Applications Worksheet Omega 1. Sinusoidal Functions as Mathematical Models 1. Ferris wheel 1 See the answer help with the problems. θ Write an equation about the movement of a Ferris wheel. 1 – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. If we can find a suitable . Trigonometric Functions Chapter 5 TexPoint fonts used in EMF. b) If a passenger gets on at the bottom of the wheel, when is the passenger 30 m above the Precalculus Chapter 6 Worksheet Graphing Sinusoidal Functions in Degree Mode Find the amplitude, period, phase (horizontal) displacement and translation (vertical displacement). After this minimum depth, the first maximum depth of 20m occurs at 10:30 am. The equation for the height To answer the Ferris wheel problem at the beginning of the section, we need to be able to express our sine and cosine functions at inputs of time. The highest point on the wheel is 43 feed above the ground. In this section, we will interpret and create graphs of sine and cosine functions. Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions (5. The answers to the even-numbered problems and to the “Additional Problems” problems are TI-83 plus directions hyperbola, Ti-84 plus spiele downloaden, Solving Quadratic Functions by square root property, how to do solve equation for free on line, real-life examples of hyperbolas. The ride starts at the bottom. the bottom of the wheel is 1. Ferris Wheel - Jim King, University of Washington Physical devices can be modeled using dynamic geometry. : AAAAAAAA Angles and Their Measure Section 5. I honestly have no idea how to begin. The international standard Mr. y = 7+4cos3(q-10°) ©e F2y0Q1y6U YKQultraD LS_oyfqt]wYaCroel FLzLVCo. 6. Planned to stand at 208 m (680 ft), significantly higher than both the current A population of animals oscillates sinusoidally between a low of 400 on January 1 and a high of 1400 on July 1. 5. (t=0 when you entered). Draw an appropriate diagram illustrating the details of this problem a. 8(x + 1. Midline = b. Graph the population against time and use your graph to find a formula for the population P as a function of time t, in months since the start of the year. If so, take a look at the procedure for this in the Modeling Real World Problems section. Then consider a ride on the Singapore Flyer, the world's tallest Ferris wheel. A seat’s position on a Ferris wheel can be modelled by the function y = 18 cos 2. Imagine a bicycle, wheel whose radius is one unit, with a marker attached to the rim of the rear wheel, as shown in the following figure. pdf file. In a predator/prey model, the predator population is modeled by the function: 2 900cos 8000 3 pt §·S ¨¸ ©¹ where t is measured in years. That means that if you were to graph the position of the seat, you could create a circle centered at the origin. When the last seat is filled, your seat is somewhere on the right side of the wheel. 69 Use the data to make defensible predictions about the characteristics of this function in terms of a sinusoidal function and use those characteristics to write the equation of the function in terms You can use trigonometry to graph the changes in high and low tides for a particular location. Ferris Wheel Problem. Math sinusoidal function problem? A ferris wheel has a diameter of 20m and is 4m above Ground level at it's lowest point. Trigonometry Word Problems Worksheet - Answers. Rocket Rides to design a Ferris wheel, given the following constraints. If we can find a suitable Math 11 Test Outline Sinusoidal Models 5. When the last seat is filled and the Ferris wheel starts, your seat is at the position shown in 6aabove. When the last seat is filled, and the Ferris wheel starts, you are in the position indicated in the diagram below: So we were given an in-class assignment to work on for 15 minutes as a group and come up with the answers. A platform allows a passenger to get on the Ferris wheel at a point P which is 20m above the ground. Instead of doing a textbook problem with a fictional Ferris wheel, I decided to use a real Ferris wheel from a nearby amusement park that some of my students would be familiar with. How to use SOHCAHTOA to calculate the height of trees, buildings etc. amplitude, frequency, period, and critical points). So now we are ready to dive into applications of sinusoidal functions. Using a stopwatch, you begin when you are at the bottom, 3 feet above the ground. ferris wheel problem sinusoidal functions answers

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