Ferris Wheel
Ferris Wheel Al and Betty have to go the amusement park to ride on a Ferris wheel. The wheel in the park has a radius of 15 feet, and its center is 20 feet above ground level. You can describe various positions in the cycle of the Ferris wheel in terms of the face of a clock, as indicated in the accompanying diagram. For example, the highest point in the wheel’s cycle is the 12 o’clock position, and the point farthest to the right is the 3 o’clock position. For simplicity, think of Al and Betty’s location as they ride as simply a point on the circumference of the wheel’s circular path. That is, ignore the size of the Ferris wheel seats, Al and Betty’s own heights, and so on. 1) How far off the ground are Al and Betty when they are at each of the following positions? a) the 3:00 position b) the 12:00 position c) the 9:00 position d) the 6:00 position 2) How far off the ground are Al and Betty when they are at the 2:00 position?
3) Pick two other clock positions and Nigure out how far off the ground Al and Betty are when they reach each of these positions
As The Ferris Wheel Turns The Ferris wheel described in the previous problem completes a rotation in 40 seconds and turns in a counterclockwise direction. And remember, that C = 2Πr. 1) At what speed are Al and Betty moving (in feet per second) as they go around the on the Ferris wheel?
2) Through what angle (in degrees) does the Ferris wheel turn each second? (The rate at which an object turns is called angular speed, because it measures how fast an angle is changing. Angular speed does not depend on the radius.)
3) How many seconds does it take for the platform to go each of these distances? a) from 3:00 to 11:00 b) from 3:00 to 7:00 c) from 3:00 to 4:00 4) What is Al and Betty’s height off the ground at each of these times? a) 1 second after passing the 3:00 position b) 6 seconds after passing the 3:00 position c) 10 seconds after passing the 3:00 position d) 14 seconds after passing the 3:00 position e) 23 seconds after passing the 3:00 position f) 49 seconds after passing the 3:00 position
5) Write a general formula that shows their height as a function of time since they went past the 3:00 position.
A Clear View The Ferris wheel now turns with a constant angular speed and takes 24 seconds for a complete turn. There is a 13-‐foot fence around the amusement park, but once you get above the fence, there is a wonderful view. 1) What percentage of the time are Al and Betty above the height of the fence? (You may want to Nind out how long they are above the height of the fence during each complete turn of the Ferris wheel.)
2) How would the answer to Question 1 change if the period were different from 24 seconds?
Graphing the Ferris Wheel 1) Plot individual points to create a graph showing Al and Betty's height, h, as a function of the time elapsed, t. Your graph should show the Nirst 80 seconds of the Ferris wheel's movement. (Assume the Ferris wheel takes 40 seconds to make a compete rotation.)
2) Describe in words how this graph would changeif you made each of the changes described below. Treat each question as a separate problem, changing only the item mentioned in that problem and keeping the rest of the information as in Question 1. a) How would the graph change if the radius of the Ferris wheel were smaller?
b) How would the graph change if the Ferris wheel was turning faster (that is, if the period was shorter)?
c) How would the graph change if you measured height with respect to the center of the Ferris wheel instead of with respect to the ground? (For example, Al and Betty 15 feet off the ground, you would treat this as a height of -‐5 beacuse 15 feet above the ground is 5 feet below the center of the Ferris wheel.)
Sand Castles Shelly loves to build elaborate sand castles at the beach. Her big problem is that her sand castles take a long time to build, and they often get swept away by the in-‐coming tide. Shelly is planning another trip to the beach next week. She decides to pay attention to the tides so that she can plan her castle-‐building and have as much time as possible. The beach slopes gradually up from the ocean toward the parking lot. Shelly considers the waterline to be “high” if the water comes up farther up the beach, leaving less sandy area available. She considers the waterline to be “low” if there is more sandy area visible on the beach. Shelly likes to build as close to the water as possible because the damp sand is better for building. According to Shelly’s analysis, the level of the water on the beach for the day of her trip will Nit the equation:
w(t) = 20sin 29t In this equation, w(t) represents how far the waterline is above or below its average position. The distance is measured in feet, and t represents the number of hours elapsed since midnight. In the case shown in the accompanying diagram, the water has come up above its average position, and w(t) is positive.
Sand Castles 1) Graph the waterline function for a 24-‐hour period. 2) a. What is the highest up on the beach (compared to the average position) that the waterline will be during the day? (This is called high tide, for you landlubbers.) b. What is the lowest that the waterline will be during the day? (This is called low tide.) 3) Suppose Shelly plans to build her castle right on the average waterline just as the water has moved below that line. How much time will she have to build her castle before the water returns and destroys her work? 4) Suppose Shelly wants to build her castle 10 feet below the average waterline. What is the maximum amount of time she can arrange to have to make her castle? Suppose Shelly decides she needs only two hours to build and admire her castle. What is the lowest point on the beach where she can build it?
More Beach Adventures 1) After spending some of the day at the beach building sand castles, Shelly wants to take an evening with a friend along the shoreline (no, really, they’re “just friends.”) Shelly knows that at one place along the shore, it is quite rocky. At that point, the rocks jut into t he ocean so that in order to pass around them, a person has to walk along a path that is 14 feet below the average waterline. Assume that Shelly and her friend don’t want tot get their feet wet. Therefore, they need to take their walk during the time when the waterline is 14 feet or more below the average waterline. What is the time period during which they can take their walk? (Recall that the position of the waterline over the course of the day is given by the equation w(t) = 20sin 29t , where the distance is measured in feet and t represents the number of hours elapsed since midnight.) 2) Shelly often Ninds herself looking for number whose sine is a given value (don’t you?). This question asks you to do the same. Your solutions should all be between -‐360o and 360o. In 2a, 2b, and 2c Nind exact values for θ. In 2d, give θ to the nearest degree. a. Find three values of θ, other than 15o, such that sin θ = sin 15o. b. Find three values of θ such that sin θ = sin 60o. c. Find three values of θ such that sin θ = 0.5. d. Find three values of θ such that sin θ = -‐0.71.
Generalizing the Ferris Wheel If Al and Betty start at the 3:00 position, with the Ferris wheel turning at counterclockwise at a constant angular speed of 9 degrees per second, then they will remain in the Nirst quadrant through t = 10. During this time interval, their x-‐coordinate is given by the formula x = 50 cos 9t. This formula speciNically uses the fact that the radius is 50 feet and that the angular speed is 9 degrees per second. But the right-‐triangle deNinition of the cosine function only applies to acute angles, so this formula isn't deNined if t is greater than 10. Your task in this activity is to explore how to extend the deNinition of the cosine function.
1) Consider the case t = 12. a. Find Al and Betty's x-‐coordinate when t = 12. Express your answer in terms of the cosine of some acute angle. b. What value should you assign to cos (9 ⋅ 12) so that the formula x = 50 cos 9t gives your answer from question 1a when you substitute 12 for t? 2) Consider the case t = 26. a. Find Al and Betty's x-‐coordinate when t = 26. Express your answer in terms of the cosine of some acute angle. b. What value should you assign to cos (9 ⋅ 26) so that the formula x = 50 cos 9t gives your answer from Question 1a when you substitute 26 for t? 3) How can you deNine cos θ in a way that makes sense for all angles and gives the results you needed in Questions 1b and 2b (or not 2b!)?
What's Your Cosine? You have seen that we deNine the cosine function in a manner similar to that for the sine function. If θ is any angle, we draw a ray from the origin, making a counterclockwise angle of that size with the positive x-‐axis, pick a point (x, y) on the ray (other than the origin), and deNine r as the distance from (x, y) to the origin, so r = √(x2 + y2). We then deNine the cosine function for all angles by the equation cos θ = x/r As with the sine function, this deNinition gives the same values for acute angles as the right-‐triangle deNinition. Also, like the sine function, the extended cosine function can give the same value for different angles. 1) Draw the graph of the function deNined by the equation z = cos θ for values of θ from -‐360o to 720o, and answer these questions. a. What is the amplitude of this function? b. What is the period of this function? Why is the cosine periodic? c. What are the θ-‐intercepts of the graph? d. What values of θ make cos θ a maximum? What values of θ make cos θ a minimum?
2) a. Find three values of θ, other than 81o, such that cos θ = cos 81o. b. Find three values of θ, such that cos θ = -‐cos 20o. c. Find three values of θ, such that cos θ = 0.3. d. Find three values of θ, such that cos θ = -‐0.48.
Find the Ferris Wheel 1) Imagine that the equations in Questions 1a and 1b are each though of as describing the x-‐coordinate of a rider on some Ferris wheel in terms of time, where the rider is at the 3:00 position when t = 0. (Here, t is in seconds and x is in feet.) Give the radius, period, and angular speed of the Ferris wheel that each expression represents. (Recall that angular speed is the rate at which the Ferris wheel turns and in this situation is given in degrees per second.) a. x = 25 cos 10t b. x = 100 cos 3t 2) a. Write an expression that would give the x-‐coordinate of a rider on a Ferris wheel that has a smaller radius than the Ferris wheel in Question 1a but a greater angular speed. b. Describe how the graph for the expression in Question 2a would differ from the graph in Question 1a.
3) Pick two other clock positions and Nigure out how far off the ground Al and Betty are when they reach each of these positions
As The Ferris Wheel Turns The Ferris wheel described in the previous problem completes a rotation in 40 seconds and turns in a counterclockwise direction. And remember, that C = 2Πr. 1) At what speed are Al and Betty moving (in feet per second) as they go around the on the Ferris wheel?
2) Through what angle (in degrees) does the Ferris wheel turn each second? (The rate at which an object turns is called angular speed, because it measures how fast an angle is changing. Angular speed does not depend on the radius.)
3) How many seconds does it take for the platform to go each of these distances? a) from 3:00 to 11:00 b) from 3:00 to 7:00 c) from 3:00 to 4:00 4) What is Al and Betty’s height off the ground at each of these times? a) 1 second after passing the 3:00 position b) 6 seconds after passing the 3:00 position c) 10 seconds after passing the 3:00 position d) 14 seconds after passing the 3:00 position e) 23 seconds after passing the 3:00 position f) 49 seconds after passing the 3:00 position
5) Write a general formula that shows their height as a function of time since they went past the 3:00 position.
A Clear View The Ferris wheel now turns with a constant angular speed and takes 24 seconds for a complete turn. There is a 13-‐foot fence around the amusement park, but once you get above the fence, there is a wonderful view. 1) What percentage of the time are Al and Betty above the height of the fence? (You may want to Nind out how long they are above the height of the fence during each complete turn of the Ferris wheel.)
2) How would the answer to Question 1 change if the period were different from 24 seconds?
Graphing the Ferris Wheel 1) Plot individual points to create a graph showing Al and Betty's height, h, as a function of the time elapsed, t. Your graph should show the Nirst 80 seconds of the Ferris wheel's movement. (Assume the Ferris wheel takes 40 seconds to make a compete rotation.)
2) Describe in words how this graph would changeif you made each of the changes described below. Treat each question as a separate problem, changing only the item mentioned in that problem and keeping the rest of the information as in Question 1. a) How would the graph change if the radius of the Ferris wheel were smaller?
b) How would the graph change if the Ferris wheel was turning faster (that is, if the period was shorter)?
c) How would the graph change if you measured height with respect to the center of the Ferris wheel instead of with respect to the ground? (For example, Al and Betty 15 feet off the ground, you would treat this as a height of -‐5 beacuse 15 feet above the ground is 5 feet below the center of the Ferris wheel.)
Sand Castles Shelly loves to build elaborate sand castles at the beach. Her big problem is that her sand castles take a long time to build, and they often get swept away by the in-‐coming tide. Shelly is planning another trip to the beach next week. She decides to pay attention to the tides so that she can plan her castle-‐building and have as much time as possible. The beach slopes gradually up from the ocean toward the parking lot. Shelly considers the waterline to be “high” if the water comes up farther up the beach, leaving less sandy area available. She considers the waterline to be “low” if there is more sandy area visible on the beach. Shelly likes to build as close to the water as possible because the damp sand is better for building. According to Shelly’s analysis, the level of the water on the beach for the day of her trip will Nit the equation:
w(t) = 20sin 29t In this equation, w(t) represents how far the waterline is above or below its average position. The distance is measured in feet, and t represents the number of hours elapsed since midnight. In the case shown in the accompanying diagram, the water has come up above its average position, and w(t) is positive.
Sand Castles 1) Graph the waterline function for a 24-‐hour period. 2) a. What is the highest up on the beach (compared to the average position) that the waterline will be during the day? (This is called high tide, for you landlubbers.) b. What is the lowest that the waterline will be during the day? (This is called low tide.) 3) Suppose Shelly plans to build her castle right on the average waterline just as the water has moved below that line. How much time will she have to build her castle before the water returns and destroys her work? 4) Suppose Shelly wants to build her castle 10 feet below the average waterline. What is the maximum amount of time she can arrange to have to make her castle? Suppose Shelly decides she needs only two hours to build and admire her castle. What is the lowest point on the beach where she can build it?
More Beach Adventures 1) After spending some of the day at the beach building sand castles, Shelly wants to take an evening with a friend along the shoreline (no, really, they’re “just friends.”) Shelly knows that at one place along the shore, it is quite rocky. At that point, the rocks jut into t he ocean so that in order to pass around them, a person has to walk along a path that is 14 feet below the average waterline. Assume that Shelly and her friend don’t want tot get their feet wet. Therefore, they need to take their walk during the time when the waterline is 14 feet or more below the average waterline. What is the time period during which they can take their walk? (Recall that the position of the waterline over the course of the day is given by the equation w(t) = 20sin 29t , where the distance is measured in feet and t represents the number of hours elapsed since midnight.) 2) Shelly often Ninds herself looking for number whose sine is a given value (don’t you?). This question asks you to do the same. Your solutions should all be between -‐360o and 360o. In 2a, 2b, and 2c Nind exact values for θ. In 2d, give θ to the nearest degree. a. Find three values of θ, other than 15o, such that sin θ = sin 15o. b. Find three values of θ such that sin θ = sin 60o. c. Find three values of θ such that sin θ = 0.5. d. Find three values of θ such that sin θ = -‐0.71.
Generalizing the Ferris Wheel If Al and Betty start at the 3:00 position, with the Ferris wheel turning at counterclockwise at a constant angular speed of 9 degrees per second, then they will remain in the Nirst quadrant through t = 10. During this time interval, their x-‐coordinate is given by the formula x = 50 cos 9t. This formula speciNically uses the fact that the radius is 50 feet and that the angular speed is 9 degrees per second. But the right-‐triangle deNinition of the cosine function only applies to acute angles, so this formula isn't deNined if t is greater than 10. Your task in this activity is to explore how to extend the deNinition of the cosine function.
1) Consider the case t = 12. a. Find Al and Betty's x-‐coordinate when t = 12. Express your answer in terms of the cosine of some acute angle. b. What value should you assign to cos (9 ⋅ 12) so that the formula x = 50 cos 9t gives your answer from question 1a when you substitute 12 for t? 2) Consider the case t = 26. a. Find Al and Betty's x-‐coordinate when t = 26. Express your answer in terms of the cosine of some acute angle. b. What value should you assign to cos (9 ⋅ 26) so that the formula x = 50 cos 9t gives your answer from Question 1a when you substitute 26 for t? 3) How can you deNine cos θ in a way that makes sense for all angles and gives the results you needed in Questions 1b and 2b (or not 2b!)?
What's Your Cosine? You have seen that we deNine the cosine function in a manner similar to that for the sine function. If θ is any angle, we draw a ray from the origin, making a counterclockwise angle of that size with the positive x-‐axis, pick a point (x, y) on the ray (other than the origin), and deNine r as the distance from (x, y) to the origin, so r = √(x2 + y2). We then deNine the cosine function for all angles by the equation cos θ = x/r As with the sine function, this deNinition gives the same values for acute angles as the right-‐triangle deNinition. Also, like the sine function, the extended cosine function can give the same value for different angles. 1) Draw the graph of the function deNined by the equation z = cos θ for values of θ from -‐360o to 720o, and answer these questions. a. What is the amplitude of this function? b. What is the period of this function? Why is the cosine periodic? c. What are the θ-‐intercepts of the graph? d. What values of θ make cos θ a maximum? What values of θ make cos θ a minimum?
2) a. Find three values of θ, other than 81o, such that cos θ = cos 81o. b. Find three values of θ, such that cos θ = -‐cos 20o. c. Find three values of θ, such that cos θ = 0.3. d. Find three values of θ, such that cos θ = -‐0.48.
Find the Ferris Wheel 1) Imagine that the equations in Questions 1a and 1b are each though of as describing the x-‐coordinate of a rider on some Ferris wheel in terms of time, where the rider is at the 3:00 position when t = 0. (Here, t is in seconds and x is in feet.) Give the radius, period, and angular speed of the Ferris wheel that each expression represents. (Recall that angular speed is the rate at which the Ferris wheel turns and in this situation is given in degrees per second.) a. x = 25 cos 10t b. x = 100 cos 3t 2) a. Write an expression that would give the x-‐coordinate of a rider on a Ferris wheel that has a smaller radius than the Ferris wheel in Question 1a but a greater angular speed. b. Describe how the graph for the expression in Question 2a would differ from the graph in Question 1a.
