(Modeling) Stopping Distance on a Curve Refer to Exercise

When bearing is given as a single angle measure, how is the angle represented in a sketch?

When bearing is given as N (or S), then an acute angle measure, and then E (or W), how is the angle represented in a sketch?

Why is it important to draw a sketch before solving trigonometric problems like those in the last two sections of this chapter?

How should the angle of elevation (or depression) from a point X to a point Y be represented?

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 1-4, 02

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 15, 02

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 10, 42

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 10, -22

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 1-5, 52

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 1-3, -32

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 12, -22

Concept Check An observer for a radar station is located at the origin of a coordinate system. For each of the points in Exercises 5 12, find the bearing of an airplane located at that point. Express the bearing using both methods. 12, 22 200

The ray y = x, x 0, contains the origin and all points in the coordinate system whose bearing is 45. Determine the equation of a ray consisting of the origin and all points whose bearing is 240.

Repeat Exercise 13 for a bearing of 150.

Distance Flown by a Plane A plane flies 1.3 hr at 110 mph on a bearing of 38. It then turns and flies 1.5 hr at the same speed on a bearing of 128. How far is the plane from its starting point?

Distance Traveled by a Ship A ship travels 55 km on a bearing of 27 and then travels on a bearing of 117 for 140 km. Find the distance from the starting point to the ending point.

Distance between Two Ships Two ships leave a port at the same time. The first ship sails on a bearing of 40 at 18 knots (nautical miles per hour) and the second on a bearing of 130 at 26 knots. How far apart are they after 1.5 hr?

Distance between Two Ships Two ships leave a port at the same time. The first ship sails on a bearing of 52 at 17 knots and the second on a bearing of 322 at 22 knots. How far apart are they after 2.5 hr?

Distance between Two Docks Two docks are located on an east-west line 2587 ft apart. From dock A, the bearing of a coral reef is 58 22. From dock B, the bearing of the coral reef is 328 22. Find the distance from dock A to the coral reef.

Distance between Two Lighthouses Two lighthouses are located on a north-south line. From lighthouse A, the bearing of a ship 3742 m away is 129 43. From lighthouse B, the bearing of the ship is 39 43. Find the distance between the lighthouses.

Distance between Two Ships A ship leaves its home port and sails on a bearing of S 61 50 E . Another ship leaves the same port at the same time and sails on a bearing of N 28 10 E . If the first ship sails at 24.0 mph and the second sails at 28.0 mph, find the distance between the two ships after 4 hr.

Distance between Transmitters Radio direction finders are set up at two points A and B, which are 2.50 mi apart on an east-west line. From A, it is found that the bearing of a signal from a radio transmitter is N 36 20 E , and from B the bearing of the same signal is N 53 40 W. Find the distance of the transmitter from B.

Flying Distance The bearing from A to C is S 52 E. The bearing from A to B is N 84 E. The bearing from B to C is S 38 W. A plane flying at 250 mph takes 2.4 hr to go from A to B. Find the distance from A to C.

Flying Distance The bearing from A to C is N 64 W. The bearing from A to B is S 82 W. The bearing from B to C is N 26 E. A plane flying at 350 mph takes 1.8 hr to go from A to B. Find the distance from B to C.

Distance between Two Cities The bearing from Winston-Salem, North Carolina, to Danville, Virginia, is N 42 E. The bearing from Danville to Goldsboro, North Carolina, is S 48 E. A car driven by Ellen Winchell, traveling at 65 mph, takes 1.1 hr to go from Winston-Salem to Danville and 1.8 hr to go from Danville to Goldsboro. Find the distance from Winston-Salem to Goldsboro.

Distance between Two Cities The bearing from Atlanta to Macon is S 27 E, and the bearing from Macon to Augusta is N 63 E. An automobile traveling at 62 mph needs 1 14 hr to go from Atlanta to Macon and 1 34 hr to go from Macon to Augusta. Find the distance from Atlanta to Augusta.

Solve the equation ax = b + cx for x in terms of a, b, and c. (Note: This is essentially the calculation carried out in Example 4)

Explain why the line y = 1tan u21x - a2 passes through the point 1a, 02 and makes an angle u with the x-axis.

Find the equation of the line passing through the point 125 , 02 that makes an angle of 35 with the x-axis.

Find the equation of the line passing through the point 15 , 02 that makes an angle of 15 with the x-axis.

Find h as indicated in the figure.

Find h as indicated in the figure.

Height of a Pyramid The angle of elevation from a point on the ground to the top of a pyramid is 35 30. The angle of elevation from a point 135 ft farther back to the top of the pyramid is 21 10. Find the height of the pyramid. 35 30_ 21 10_ h 135 ft

Distance between a Whale and a Lighthouse Debbie Glockner-Ferrari, a whale researcher, is watching a whale approach directly toward a lighthouse as she observes from the top of this lighthouse. When she first begins watching the whale, the angle of depression to the whale is 15 50. Just as the whale turns away from the lighthouse, the angle of depression is 35 40. If the height of the lighthouse is 68.7 m, find the distance traveled by the whale as it approached the lighthouse.

Height of an Antenna A scanner antenna is on top of the center of a house. The angle of elevation from a point 28.0 m from the center of the house to the top of the antenna is 27 10, and the angle of elevation to the bottom of the antenna is 18 10. Find the height of the antenna.

Height of Mt. Whitney The angle of elevation from Lone Pine to the top of Mt. Whitney is 10 50. Van Dong Le, traveling 7.00 km from Lone Pine along a straight, level road toward Mt. Whitney, finds the angle of elevation to be 22 40. Find the height of the top of Mt. Whitney above the level of the road.

(Modeling) Distance between Two Points Refer to Example 3. A variation of the subtense bar method that surveyors use to determine larger distances d between two points P and Q is shown in the figure. In this case the subtense bar with length b is placed between the points P and Q so that the bar is centered on and perpendicular to the line of sight connecting P and Q. The angles a and b are measured from points P and Q, respectively. (Source: Mueller, I. and K. Ramsayer, Introduction to Surveying, Frederick Ungar Publishing Co.) (a) Find a formula for d involving a , b , and b. (b) Use your formula to determine d if a = 37 48, b = 42 03, and b = 2.000 cm.

Height of a Plane above Earth Find the minimum height h above the surface of Earth so that a pilot at point A in the figure can see an object on the horizon at C, 125 mi away. Assume that the radius of Earth is 4.00 * 103 mi. C 125 mi A

Distance of a Plant from a Fence In one area, the lowest angle of elevation of the sun in winter is 23 20. Find the minimum distance x that a plant needing full sun can be placed from a fence 4.65 ft high.

Distance through a Tunnel A tunnel is to be built from A to B. Both A and B are visible from C. If AC is 1.4923 mi and BC is 1.0837 mi, and if C is 90, find the measures of angles A and B.

(Modeling) Highway Curves A basic highway curve connecting two straight sections of road is often circular. In the figure, the points P and S mark the beginning and end of the curve. Let Q be the point of intersection where the two straight sections of highway leading into the curve would meet if extended. The radius of the curve is R, and the central angle u denotes how many degrees the curve turns. (Source: Mannering, F. and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) (a) If R = 965 ft and u = 37 , find the distance d between P and Q. (b) Find an expression in terms of R and u for the distance between points M and N.

Length of a Side of a Piece of Land A piece of land has the shape shown in the figure at the left. Find the length x.

(Modeling) Stopping Distance on a Curve Refer to Exercise 41. When an automobile travels along a circular curve, objects like trees and buildings situated on the inside of the curve can obstruct the drivers vision. These obstructions prevent the driver from seeing sufficiently far down the highway to ensure a safe stopping distance. In the figure, the minimum distance d that should be cleared on the inside of the highway is modeled by the equation d = R a1 - cos u 2 b . (Source: Mannering, F. and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) Not to scale R R d (a) It can be shown that if u is measured in degrees, then u _ 57.3S R , where S is the safe stopping distance for the given speed limit. Compute d to the nearest foot for a 55 mph speed limit if S = 336 ft and R = 600 ft. (b) Compute d to the nearest foot for a 65 mph speed limit given S = 485 ft and R = 600 ft. (c) How does the speed limit affect the amount of land that should be cleared on the inside of the curve?

(Modeling) Distance of a Shot Put A shot-putter trying to improve performance may wonder whether there is an optimal angle to aim for, or whether the velocity (speed) at which the ball is thrown is more important. The figure shows the path of a steel ball thrown by a shot-putter. The distance D depends on initial velocity v, height h, and angle u when the ball is released. D _ One model developed for this situation gives D as D = v2 sin u cos u + v cos u 21v sin u22 + 64h 32 . Typical ranges for the variables are v: 3346 ft per sec; h: 68 ft; and u: 4045. (Source: Kreighbaum, E. and K. Barthels, Biomechanics, Allyn & Bacon.) (a) To see how angle u affects distance D, let v = 44 ft per sec and h = 7 ft. Calculate D, to the nearest hundredth, for u = 40 , 42, and 45. How does distance D change as u increases? (b) To see how velocity v affects distance D, let h = 7 and u = 42 . Calculate D, to the nearest hundredth, for v = 43 , 44, and 45 ft per sec. How does distance D change as v increases? (c) Which affects distance D more, v or u? What should the shot-putter do to improve performance?