I Was Going How Fast? Accident investigators use the relationship s = 21d to determine the approximate speed of a car, s mph, from a skid mark of length d feet, that it leaves during an emergency stop. This formula assumes a dry road surface and average tire wear. 1. A police officer investigating an accident finds a skid mark 115 feet long. Approximately how fast was the car going when the driver applied the brakes? 2. If a car is traveling at 60 mph and the driver applies the brakes in an emergency situation, how much distance does your model say is required for the car to come to a complete stop? 3. What is a realistic domain and range for this situation? 4. Does doubling the length of the skid double the speed the driver was going? Justify your response using tables, symbols, and graphs. Chapter 5: Square Root Functions 227 I Was Going How Fast? Teacher Notes Notes Scaffolding Questions: Materials: • Would the relationship between a skid mark and the speed the car was going when it began to stop be a functional relationship? • What factors influence realistic domain and range values for the situation? • What is involved in stopping a car? Is it more than what a skid mark shows? • Can a car stop without leaving a skid mark? • Besides speed, what else would contribute to the length of a skid mark? • Did the function that was given relating skid mark to speed take any other conditions into account? • What might you change about the equation to take some of these other elements into account? Graphing calculator Algebra II TEKS Focus: (2A.9) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: (C) determine the reasonable domain and range values of square root functions, as well as interpret and determine the reasonableness of solutions to square root equations and inequalities. (D) determine solutions of square root equations using graphs, tables, and algebraic methods. (F) analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems. Additional Algebra II TEKS: (2A.1) Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. 228 Sample Solutions: 1. The given value of 115 feet is the length of a skid mark, d. Substitute for d in the formula. s = 21d s = 21* 115 s ≈ 49 mph 2. The given value of 60 mph is the approximate speed of the car, s. Substitute for s in the formula and solve for d. s = 21d 60 = 21d 3600 = 21d d ≈ 171 feet 3. The function has a domain and a range of all positive real numbers. The situation dictates that there is some realistic maximum length and speed. Chapter 5: Square Root Functions I Was Going How Fast? Teacher Notes 4. Doubling the length of the skid mark does not double the speed. The table below shows that when the skid value is doubled the speed value is not doubled. For example, 50 = 2(25) but 32.4 is not twice 22.9. skid speed 25 ft 50 ft 100 ft 200 ft 400 ft 22.9 mph 32.4 mph 45.8 mph 64.8 mph 91.7 mph The graph also shows the function is not linear. Notice the labeled points. The student is expected to: (A) identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations. (2A.9) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. To show this symbolically let s’ represent the speed when the skid mark length is doubled. s ' = 21 * (2d ) s ' = 2 21d s ' = 1.414 21d The speed is changed by a factor of approximately 1.414. It is not doubled. Extension Questions: • For what type of function would doubling the length of the x-value double the y-value? Linear functions of the form y = mx would be such that if x is doubled then y will be doubled. 2y = m(2x) Chapter 5: Square Root Functions (B) relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions. Connection to TAKS: Objective 2: The student will demonstrate an understanding of the properties and attributes of functions. Objective 5: The student will demonstrate an understanding of quadratic and other nonlinear functions. Objective 10: The student will demonstrate an understanding of the mathematical processes and tools used in problem solving. 229 I Was Going How Fast? Teacher Notes • Express the length of a skid mark as a function of speed. s = 21d s 2 = 21d d = s2 21 Note that the square root function and this quadratic function are inverses of each other for positive values of s. • Investigators find that a car that caused an accident left a skid mark 143 feet long. Damage to the car reveals that it was moving at a rate of 30 mph when it hit the other car. How fast was the car going when it started to skid? If the skidding car had not been stopped by hitting another car, it would have needed 302 another 21 ≈ 42.8 feet to skid before stopping. The total skid would have been (143 + 43) feet. A skid mark this long would imply the car was going 21* 186 ≈ 62.5 mph when it started to skid. • What, besides the actual braking distance, do you think affects the total distance that it takes to stop a car in an emergency? The distance that the car travels while the driver is reacting to the emergency situation needs to be added to the braking distance to determine the total stopping distance. • There is a building on a corner of the highway that blocks a driver’s view for 150 feet. If the speed limit on this stretch of highway is 55 mph, does a driver have enough time to stop if there is a car broken down in the highway 150 feet around the corner? Determine the distance to stop at 55 mph hour using the rule. s = 21d 55 = 21d 552 = 21d 3025 ≈ 144 feet d = 21 Assuming the car could begin to stop the instant the driver saw the broken-down car, it would take 144 feet to stop from a speed of 55 mph. However, other factors that are not taken into account here, such as a driver’s reaction time, would increase this stopping distance. Therefore, there is probably not enough room. 230 Chapter 5: Square Root Functions

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