10/20/15 ALICATIONS DAY #3 HOMEWORK TC2 WARM U! Agenda Homework - go over homework #2 on applications - Finish Applications Day #3 - more applications... tide problems, start project UCOMING: OW #6 Quiz Friday 10/23 with Quiz #4 Unit 2 Quiz #4 on Applications Friday 10/23 HUGE UNIT 2 TEST!! Tuesday 10/27 1. f (x) = 4sin π ( 3 x 3) +1 Vertical Shift: Amplitude: Horizontal Shift: Frequency: eriod: Critical values happen every: Critical values are: Now, Graph the sinusoidal function. Label your axes clearly. 2. Evaluate the following: A. cos 330 ( ) = B. sin 7π 4 =
ALICATIONS OF SINUSOIDAL FUNCTIONS DAY #3 1. Tidal River: In a tidal river, the time between high tide and low tide is approximately 6.2 hours. The average depth of the water in a port on the river is 4 meters. High tide is at 1:15 AM and, the depth is 5 meters. a. Graph the water in the port over time if the relationship between the time and depth is sinusoidal. Let x represent the number of hours after 12:00 midnight. (, ) and (, _) VS = A = HS = = F = 2π c. Cosine Equation: d. How deep is the water at 2:00pm? Round to three decimal places.
2. The Ferris Wheel roblem: As you ride a Ferris wheel, your distance from the ground varies sinusoidally with time. When the last seat is filled and the Ferris wheel starts. Let t be the number of seconds that have elapsed since the Ferris wheel started. You find that it takes 3 seconds to reach the top, 43 feet above the ground. The wheel makes a revolution once every 8 seconds. The diameter of the wheel is 40 feet. a. Sketch a graph of this sinusoid. High points: (, ) and (, ) Low point: (, ) b. What is the lowest you go as the Ferris wheel turns, and why is this number greater than zero? c. Find the vertical shift, amplitude, horizontal shift, period, and frequency. VS = A = HS = = F = 2π d. Cosine Equation: e. Find your height above the ground when: (round to the nearest tenth) t = 6 seconds h = t = 4 seconds h = t = 9 seconds h = t = 0 seconds h = f. When do you reach a height of 21 feet within the first 12 seconds? Find all places and round to the nearest tenth.
3. The Water Wheel: Suppose that the water wheel shown in the figure below rotates at 6 revolutions per minute (rpm). You start your stopwatch. Two seconds later, point on the rim of the wheel is at its greatest height. You are to model the distance d of the point from the surface of the water in terms of the number of seconds, t, the stop what reads. Distance varies sinusoidally with time. a. Sketch the cosine curve associated with this situation. VS = A = HS = = F = 2π c. Cosine Equation: d. How far is point from the water when t = 5.5 seconds? Round to the nearest tenth. e. Find the first 3 times when point is 8 feet above the water. Round to two decimal places.
4. Roller Coaster roblem: A portion of a roller coaster track is to be built in the shape of a sinusoid. You have been hired to calculate the lengths of the vertical timber supports to be used. a. Using the diagram above, sketch the cosine curve associated with this situation. VS = A = HS = = F = 2π c. Cosine Equation: d. How long is the vertical timber at the high point (x = 0 meters)? At x = 4 meters? At x = 32 meters? Round to two decimals places.
5. Average Temperatures: The following data represents the mean monthly temperature in Columbia, South Carolina. Month Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Temp 46 50 56 65 73 79 82 80 76 65 56 47 a. Using the data from the table, sketch the cosine curve associated with this situation. Let January be x = 1. b. Maximum Value= Minimum Value= c. Find the vertical shift, amplitude, horizontal shift, period, and frequency. VS = A = HS = = F = 2π d. Cosine Equation:
6. The depth of water at a certain point varies with the tides throughout the day. On a particular day, the high tide occurs at 4:09 a.m. with a depth of 6.2 meters. The low tide occurs at 10:57 a.m. with a depth of 3 meters. The depth of the tide depends on the time of day, so time is the independent variable (x) and the depth of the tide is the dependent variable (y). The given ordered pairs are (time after 12 AM, depth): a) Sketch a graph of this situation (, ) and (, ) VS = A = HS = = F = 2π c. Cosine Equation: d. How deep is the water at 9:30 a.m.? e. When in the morning is the water 5 m deep?
7. NYSE: On the New York Stock Exchange (NYSE), there are several indexes used to measure market trends and general stock market price movement. These indexes frequently rise and fall in a sinusoidal pattern. One such index reaches its maximum index value of 1100 in the year 2001. The next time it falls to its minimum of 600 is in 2005. Let the year 2000 represent time = 0. (, ) and (, ) a. lot the two given points and then sketch the cosine curve associated with those 2 points. VS = A = HS = = F = 2π c. Cosine Equation: d. Find the index value for 2007. e. A seller s market occurs when stock market values are high and an individual can sell shares of his/her stock for a large profit. For the stocks assessed using this index, this occurs when the index value is over 1000. When is the next time interval after 2007 over which we have a seller s market? Round to three decimal places.