time? How will changes in vertical drop of the course affect race time? How will changes in the distance between turns affect race time?

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1 Unit 1 Leon 1 Invetigation 1 Think About Thi Situation Name: Conider variou port that involve downhill racing. Think about the factor that decreae or increae the time it take to travel from top to bottom. a For downhill or lalom kiing, how will change in the length of the coure affect race time? How will change in vertical drop of the coure affect race time? How will change in the ditance between turn affect race time? b What other factor will affect downhill or lalom ki-race time? How will change in each of thoe variable increae or decreae race time? c Pick another downhill race port that interet you and think about the variable that affect race time in that port event. What change in thoe variable will increae the time to travel from top to bottom? What change will decreae the time? In previou coure, you ued table of value, graph, and ymbolic rule to repreent function relating independent and dependent variable. You recognized and decribed a variety of common pattern in thoe relationhip. In thi leon, you will develop your undertanding and kill in dealing with two pecial type of relationhip direct and invere variation. Invetigation 1 On a Roll In downhill racing on ki, led, bike, or kateboard, change in two variable may have oppoite effect on time from top to bottom. Thi make it difficult to predict the combined effect if two variable change at the ame time. However, through experimentation, you can examine the effect of change in each variable eparately and then build a model of the multivariable relation. A you work on the quetion of thi invetigation, look for anwer to thee quetion: How do coure length and teepne affect run time for a downhill race? How can the relationhip between thoe variable be expreed in ymbolic form? LESSON 1 Direct and Invere Variation 3

2 Platform Height, Ramp Length, and Ride Time Ramp are often ued in kateboarding, not jut for getting air, but alo for tarting into terrain park or treet race. Many ramp are attached to a raied platform a pictured below. 1 The height of the platform and length of the ramp affect the time it take to roll down the ramp. a. For a fixed ramp length, how do you think the time it take to ride down the ramp will change a platform height increae? b. For a fixed platform height, how do you think the time it take to ride down the ramp will change a ramp length increae? c. Suppoe that one kateboard ramp i twice a long a another ramp. What relationhip between platform height for thoe ramp do you think will allow kateboarder tarting at the top of each ramp at the ame time to reach the bottom at the ame time? To explore the effect of platform height and ramp length on the time it take to ride downhill, you can conduct an experiment deigned by Galileo over 500 year ago. To get idea about the effect of gravity, he timed trip of a ball rolling down ramp of variou height and length. You can build a ramp with a piece of V-haped wood or metal. You will find it eaier to do accurate timing if the ramp are fairly long and gently loped. Carry out the ramp experiment decribed in Problem 2 and 3 below. Divide the experimental work among the member of your cla, with each team taking one fixed ramp length or platform height to tudy. Ramp Length Platform Height 4 UNIT 1 Function, Equation, and Sytem

3 2 To ee how the time T it take a ball to roll down a ramp change a the ramp length L increae, experiment with ramp that have length varying from 3 feet to 8 feet but a platform height of 0.5 feet in each cae. Then rerun the experiment uing a platform height of 0.25 feet in each cae. a. Record the reult of each experiment in a table like thi: Ramp Length (in feet) Roll Time (in econd) at 0.5-ft Height Roll Time (in econd) at 0.25-ft Height b. To tudy the pattern relating roll time to ramp length, make plot of the (L, T) data one for each platform height. c. Examine the data pattern from the two experiment. For each platform height, decribe the relationhip between roll time and ramp length. 3 To ee how the time T it take a ball to roll down a ramp change a the platform height H increae, experiment with platform height varying from 0.25 feet to 1.5 feet but fixed ramp length of 8 feet in each cae. Then change the fixed ramp length to 4 feet and ee how roll time i related to platform height in that cae. a. Record the reult of each experiment in a table like thi: Platform Height (in feet) Roll Time (in econd) for 8-ft Ramp Roll Time (in econd) for 4-ft Ramp b. To tudy the pattern relating platform height and roll time, make plot of the (H, T) data one for each ramp length. c. Examine the data pattern from the two experiment. For a fixed ramp length, decribe the relationhip between roll time and platform height. 4 Compare the reult from your experiment in Problem 2 and 3 to your repone to the quetion of Problem 1. Dicu any urprie and try to explain why the reult make ene. Familie of Variation Pattern The data pattern and graph that how how roll time depend on ramp length and platform height may remind you of other relationhip between variable that you have een in prior mathematical tudie. 5 Decide which of the following function have table and graph pattern that: are imilar to the (ramp length, time) relationhip. are imilar to the (platform height, time) relationhip. are different from thoe relationhip. Be prepared to explain your deciion. a. The ale tax on tore purchae in Michigan i a function of the purchae price and can be calculated with the formula T = 0.06p. LESSON 1 Direct and Invere Variation 5

4 b. When a club wa planning it Halloween party at the Fun Houe, the planner figured the cot per peron uing C = _ 225, where n i n the number of club member who would attend. c. The number of ticket old to a charity baketball game i a function of the price charged with rule N = 4,000-50p. d. When a doctor or nure give an injection of medicine like penicillin, the amount of active medicine t hour later can be etimated by a function like M = 300(0.6 t ). e. The circumference of a circle i related to the radiu by the formula C = 2πr. f. When a football punt leave the kicker foot, it height above the ground at any time in it flight i given by a function like h = -16t t + 4. g. The average peed for the Daytona 500 race i a function of the time it take to complete the race with rule = _ 500 t. 6 Uing your data plot from Problem 2 and 3, experiment with function graph to find function rule that eem to be good model for the relationhip between: a. roll time T and ramp length L for each platform height you teted in Problem 2. b. roll time T and platform height H for each ramp length you teted in Problem 3. Baic Variation Pattern The ituation in thi invetigation involved a variety of function relating dependent and independent variable. Several example involved pecial pattern called direct and invere variation. 6 UNIT 1 Function, Equation, and Sytem

5 Direct Variation: If the relationhip of variable y and x can be expreed in the form: y = kx for ome contant k, then we ay that y varie directly with x or that y i directly proportional to x. The number k i called the contant of proportionality for the relationhip. The cloe connection between multiplication and diviion of number implie that if y i directly proportional to x, then _ y = k. The ymbolic x form _ y = k how that the ratio of y to x i contant, for any correponding x value of y and x. 7 Explain why the perimeter P of a quare i directly proportional to the length of a ide. a. What equation how thi direct proportionality relationhip? b. What i the contant of proportionality? c. How doe the value of P change a the value of increae teadily? How i thi pattern of change related to the contant of proportionality? 8 Identify the direct variation relationhip in Problem 5. For each: a. Explain how the value of the dependent variable change a the value of the independent variable teadily increae. b. Decribe the relationhip of the variable involved by completing a entence like thi: The variable i directly proportional to, with contant of proportionality. c. Expre the relationhip between the variable in an equivalent ymbolic form that how the contant ratio of the two variable. Invere Variation: If the relationhip of variable y and x can be expreed in the form: y = _ k for ome contant k, x then we ay that y varie inverely with x or that y i inverely proportional to x. The number k i called the contant of proportionality for the relationhip. Once again, the cloe connection between multiplication and diviion of number implie that if y i inverely proportional to x, then xy = k. The ymbolic form xy = k how that the product of y and x i contant, for any correponding value of x and y. LESSON 1 Direct and Invere Variation 7

6 9 The time t required to download a 4-megabyte muic file from an Internet muic eller i inverely proportional to the rate r at which data i tranferred to the receiving computer. a. How long will it take to download a 4-megabyte file if the tranmiion occur at a rate of 2.5 megabyte per minute? How long if the tranmiion rate i 0.8 megabyte per minute? b. How can the relationhip of t and r be expreed in ymbolic form? c. How doe the value of t change a the value of r increae teadily? How i thi pattern of change related to the contant of proportionality? 10 Identify the invere variation relationhip in Problem 5. For each: a. Explain how the value of the dependent variable change a the value of the independent variable increae teadily. b. Expre the relationhip between the variable in two different but equivalent ymbolic form. c. Decribe the relationhip of the variable involved by completing a entence like thi: The variable i inverely proportional to, with contant of proportionality. 11 Examine the table below, each of which decribe a relation between x and y. Table I x y Table II x y Table III x y a. Which relation involve direct variation? What i the contant of proportionality in each cae? b. Which relation involve invere variation? What i the contant of proportionality in each cae? 8 UNIT 1 Function, Equation, and Sytem