When interpreting a word problem, graphing the situation, and writing a cosine and sine equation to model the data, use the following steps:

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Jeffery Underwood
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1 Modeling with Sinusoidal Functions Name Date PD When interpreting a word problem, graphing the situation, and writing a cosine and sine equation to model the data, use the following steps: 1) Identify ALL the information given in the problem: minimum, maximum, length of the period (or length of half the period), b using the period, amplitude (a), vertical shift (d), phase shift (c). amplitude = 2 1 (maximum minimum); radius = amplitude; 2 1 diameter = amplitude Key Points to Remember 1 period = distance from a maximum to a minimum; double this to get the period period or B B period ; one revolution = circumference = the period vertical shift (d) = (max amplitude) OR (min + amplitude) vertical shift is the movement of the midline up or down phase shift: cosine starts at a maximum, sine starts at the midline point summary divides one cycle into ' s ; one period should be at least 4 or 8 blocks 2) Graph at least two cycles of each situation, using the graph to identify any additional information needed for the equations. 3) Write a cosine AND a sine equation to model the situation. When applying an equation, generally select the cosine equation. 4) Use your model to answer any questions. When given the independent (x) value, just substitute into the equation and use your calculator to solve. When given the dependent (y) value, determine the independent (x) variable using your graph. Later we will solve these algebraically. Use the above steps to graph the following situations, write a cosine AND a sine equation, and then use your cosine model and graph to answer the questions. Use your own graph paper or the graph paper on the back of this ditto. 1) The height above the water surface y of a point on a water wheel varies sinusoidally as a function of time t. Relaxing on his back porch, Justin watches a point on a water wheel, which is 20 feet in diameter. He notices that after 4 seconds, the point is at its highest, 16 feet above the water. He also notes that the wheel makes one revolution every 24 seconds. (Hint: let the x-axis = water surface) A. Use your equation to predict the height of the point at 7, 12, 20.5 and 100 seconds. B. When is the first time the point is at the water s surface? Is it entering or coming out of the water? Explain your answer. C. When is the fourth time the point is at the water s surface?

2 2) The distance y that a swing is from a tree varies sinusoidally as a function of time t. While pushing Ashley on a swing, Candace notices that after 2 seconds, the swing is closest to the tree, at a distance of 5 feet. After 3 more seconds, the swing is farthest away from the tree, at a distance of 23 feet. (Hint: let the x-axis = tree) A. Use your equation to predict the swing s distance from the tree at 10s, 18.5s, and 2 minutes. B. When is the second time the point is 10 feet from the tree? Is it approaching or moving away from the tree? Explain your answer. 3) While parked in the city, Glenn has his tire marked at the bottom with chalk by a parking officer. When he drives off from the parking space, the height of the mark y from the ground varies sinusoidally with the horizontal distance x the tire has traveled. The diameter of the tire is 28 inches. (Hint: let the x-axis = the ground) A. Predict the height of the mark from the ground when Glenn has driven 21 inches, and the height when he has driven 100 inches. B. What are the first three horizontal distances x traveled when the mark is 25 inches above the ground.

3 Practice for Modeling For each of the following problems: A. Identify the amplitude, vertical shift, phase shift, period, and value of b. B. Sketch two cycles of the described sinusoidal graph. C. Write a cosine and sine equation for the graph. D. Answer any questions with the problem. 1) Ferris Wheel: As you ride a Ferris wheel, your distance from the ground d varies sinusoidally with time t. Let t be the number of seconds that have elapsed since the Ferris wheel started. You find that it takes you 3 seconds to reach the top of the wheel, which is 45 feet above the ground, and the wheel makes one revolution every 48 seconds. The diameter of the wheel is 40 feet. a. Use your equation to predict your height above the ground at 6, b. What is the time t the second time you are 18 feet above the ground? 4 1, 9, and 0 seconds. 3 2) Steamboat: Mark Twain sat on the deck of a river steamboat watching the paddlewheel. As the paddlewheel turned, a point on a paddle blade moved in such a way that its distance d from the water s surface was a sinusoidal function of t seconds. When his stopwatch read 4 seconds, the point was at its highest, 16 feet above the surface of the water. The wheel s diameter was 18 feet and one revolution was completed every 12 seconds. a. How far above the water s surface was the point at 5 and 17 seconds? b. When is the first positive value of t time at which the point was at the water s surface? At that time, was it entering or coming out of the water? Explain your answer.

4 3) Extraterrestrial Being: Researchers find a creature from an alien planet. Its body temperature y is varying sinusoidally with t time. They notice that 35 minutes after they beginning timing, its temperature reaches a high of 128. It reaches its next low temperature of 106 twenty minutes later. a. What is the alien s body temperature when the researchers first begin timing? b. Find the first 3 times after the researchers began timing at which the temperature was 114 F. 4) Fox Population: Naturalists find that the populations of some kinds of predatory animals vary periodically. Assume that the population p of foxes in a certain forest varies sinusoidally with time t. Records began to be kept when time t = 0. A minimum number, 200 foxes, occurred after 2.9 years. The next maximum, of 800 foxes, occurred at 5.5 years. a. Predict the population at 7 years. b. Foxes are declared to be an endangered species when their population drops below 300. Between what two nonnegative values of time t were the foxes first endangered?

5 5) Bouncing Spring: A weight attached to the end of a long spring is bouncing up and down. As it bounces, its distance d from the floor varies sinusoidally with time t. You start a stopwatch. When the stopwatch reaches 0.3 seconds, the weight first reaches a high point of 64 cm above the floor. The next low point of 40 cm above the floor occurs at 1.8 seconds. a. Predict the distance of the weight from the floor when the stopwatch reads 17.2 seconds. b. What was the distance of the weight above the floor when you started the stopwatch? c. Predict the first positive value of time at which the weight is 58 cm above the floor. 6) Pebble-in-the-Tire: As you stop your car at a traffic light, a pebble becomes wedged between the tire treads. When you start off again, the distance y of the pebble to the pavement varies sinusoidally with the distance x that you travel. The period is the circumference of the tire, and the diameter of the tire is 26 in. a. Predict the distance of the pebble from the pavement when you have traveled 15 inches. b. What are the first two distances x when the pebble is 11 inches from the pavement?

Sinusoidal Applications

Sinusoidal Applications A package of 5 activities Problems dealing with graphing and determining the equations of sinusoidal functions for real world situations Fractal image generated by MathWiz Created