563 5.3 Trigonometric Function Values and Angle Measures (Modeling) Design of Highway Curves When highway curves are designed, the outside of the curve is often slightly elevated or inclined above the inside of the curve. See the figure. This inclination is the superelevation. u For safety reasons, it is important that both the curve’s radius and superelevation be correct for a given speed limit. If an automobile is traveling at velocity V (in feet per second), the safe radius R, in feet, for a curve with superelevation u is modeled by the formula R = V2 g1ƒ + tanu2 , where ƒ and g are constants. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) 143. A roadway is being designed for automobiles traveling at 45 mph. If u = 3°, g = 32.2, and ƒ = 0.14, calculate R to the nearest foot. (Hint: 45mph = 66 ft per sec) 144. Determine the radius of the curve, to the nearest foot, if the speed in Exercise 143 is increased to 70 mph. 145. How would increasing angle u affect the results? Verify your answer by repeating Exercises 143 and 144 with u = 4°. 146. Refer to Exercise 143 and use the same values for ƒ and g. A highway curve has radius R = 1150 ft and a superelevation of u = 2.1°. What should the speed limit (in miles per hour) be for this curve? (Modeling) Measuring Speed by Radar Any offset between a stationary radar gun and a moving target creates a “cosine effect” that reduces the radar reading by the cosine of the angle between the gun and the vehicle. That is, the radar speed reading is the product of the actual speed and the cosine of the angle. (Data from Fischetti, M., “Working Knowledge,” Scientific American.) 147. Find the radar readings, to the nearest unit, for Auto A and Auto B shown in the figure. Auto A Auto B Radar gun 10° angle Actual speed: 70 mph 20° angle Actual speed: 70 mph 148. The speed reported by a radar gun is reduced by the cosine of angle u, shown in the figure, where r represents reduced speed and a represents actual speed. Use the figure to show why this “cosine effect” occurs. Auto Radar gun r a u
RkJQdWJsaXNoZXIy NjM5ODQ=