9.6 Applying Trigonometry

9-6 Applying Trigonometry 1. LIGHTHOUSES Sailors on a ship at sea spot the light from a lighthouse at an angle of elevation of 25°. The light of the lighthouse is 30 meters above sea level. How far from the shore is the ship? Round your answer to the nearest meter. SOLUTION: Let x represent the distance from the ship to the shore. The ship is 64 meters from shore. eSolutions Manual - Powered by Cognero Page 1 9-6 Applying Trigonometry 2. WATER TOWERS A student can see a water tower from the edge of a soccer field at San Lobos High School. The edge about 110 feet from the water tower, and the water tower stands at a height of 32.5 feet. What is the angle of elevation if th student viewing the tower is 6 feet above the ground? Round your answer to the nearest tenth. SOLUTION: Since the student is 6 feet tall, subtract this height from the height of the water tower, 32.5 – 6 = 26.5 feet. Let x represent t elevation. The angle of elevation is about 13.5°. 3. CONSTRUCTION A roofer props a ladder against a wall so the top of the ladder just reaches a 30-foot roof that needs repair. If the angle of elevation from the bottom of the ladder to the roof is 55°, how far is the ladder from the base of the wall? Round your answer to the nearest foot. SOLUTION: Let x represent the distance from the base of the ladder to the wall. The ladder is about 21 feet from the wall. eSolutions Manual - Powered by Cognero Page 2 9-6 Applying Trigonometry 4. MOUNTAIN BIKING On a mountain bike trip along the Gemini Bridges Trail in Moab, Utah, Nabuko stopped on the canyon floor to get a good view of the twin sandstone bridges. Nabuko is standing about 60 meters from the base of the canyon cliff, and the natural arch bridges are about 100 meters up the canyon wall. If her line of sight is 5 meters above the ground, what is the angle of elevation to the top of the bridges? Round to the nearest tenth of a degree. SOLUTION: Since Nabuko is 5 feet tall, subtract this height from the height of the natural arch bridges up the cliff, 100 – 5 = 95 feet. Let x represent the angle of elevation. The angle of elevation is about 57.7°. eSolutions Manual - Powered by Cognero Page 3 9-6 Applying Trigonometry 5. ROOFTOP Lucia is 5.5 feet tall. She is standing on the roof of a building that is 80 feet tall. She spots a fountain at ground she knows to be 122 feet away from the base of the building. What is the measure of the angle of depression formed by Lucia’s horizontal line of sight and her line of sight to the fountain? Round your answer to the nearest degree. SOLUTION: Because Lucia's horizontal line of sight and the ground are parallel, the measure of the angle made by Lucia's line of sight is angle from the ground to Lucia by the Alternate Interior Angles Theorem. Since Lucia is 5 feet tall, add this height to the height of the building, 80 + 5.5 = 85.5 feet. Let x represent the angle of depr The angle of depression is about 35°. eSolutions Manual - Powered by Cognero Page 4 9-6 Applying Trigonometry 6. AIR TRAFFIC From the top of the 120-foot-high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 19°. How far from the base of the tower is the airplane? Round to the nearest tenth of a foot. SOLUTION: Because the air traffic controller's line of sight and the runway are parallel, the angle of depression is equal to the angle from the ground to the top of the tower by the Alternate Interior Angles Theorem. The airplane is about 348.5 ft from the base of the tower. 7. AVIATION Due to a storm, a pilot flying at an altitude of 528 feet has to land. If he has a horizontal distance of 2000 feet landing strip, what angle of depression should he use to land? Round to the nearest tenth of a degree. SOLUTION: Because the horizontal flight of the plane and the ground are parallel, the measure of the angle made by the planes line of sigh strip is equal to the angle from the ground to the plane by the Alternate Interior Angles Theorem. The angle of depression is about 14.8°. eSolutions Manual - Powered by Cognero Page 5 9-6 Applying Trigonometry 8. INDIRECT MEASUREMENT Kenneth is sitting at the end of a pier and using binoculars to watch a whale surface. Th pier is 30 feet above the water, and Kenneth’s eye level is 3 feet above the pier. If the angle of depression to the whale is 20°, how far is the whale from Kenneth’s binoculars? Round your answer to the nearest tenth of a foot. SOLUTION: Because Kenneth's horizontal line of sight and the water are parallel, the measure of the angle made by Kenneth's line of sigh equal to the angle from the water to Kenneth by the Alternate Interior Angles Theorem. Since Kenneth's eye level is 3 feet above the pier, add this height to the height of the pier, 30 + 3 = 33 feet. Let x represent distance from the whale to the pier. The distance from the whale to the binoculars is about 96.5 feet. eSolutions Manual - Powered by Cognero Page 6 9-6 Applying Trigonometry 9. GARAGE To estimate the height of a garage, Carlos sights the top of the garage at a 42° angle of elevation. He then steps back 20 feet and sights the top of the garage at a 10° angle. If Carlos is 6 feet tall, how tall is the garage to the nearest foot? SOLUTION: Let Carlos' initial distance from the garage equal x and the vertical distance from Carlo's line of sight to the top of the garage equal y. Since there are two unknowns, the sides opposite and adjacent, set up a tangent ratio for both situations and solve the system of equations. Set the two equations solved for y equal to each other and solve for x, Carlos's initial distance from the garage. Substitute Carlos's initial distance into the original equation to find the vertical distance from Carlos's eye level to the top of the garage. Now add Carlos's height to get the height of the garage. 6 + 4.32 = 10.32 The garage is about 10 feet tall. 10. CLIFF Sarah stands on the ground and sights the top of a steep cliff at a 60° angle of elevation. She then steps back 50 meters and sights the top of the cliff at a 30° angle. If Sarah is 1.8 meters tall, how tall is the cliff to the nearest meter? eSolutions Manual - Powered by Cognero Page 7 9-6 Applying Trigonometry SOLUTION: Let Sarah's initial distance from the cliff equal y and the vertical distance from Sarah's line of sight to the top of the cliff equal x. Since there are two unknowns, the sides opposite and adjacent, set up a tangent ratio for both situations and solve the system of equations. Set the two equations solved for x equal to each other and solve for y, Sarah's initial distance from the cliff. Substitute Sarah's initial distance into the original equation to find the vertical distance from Sarah's eye level to the top of the cliff. Now add Sarah's height to get the height of the cliff. 1.8 + 43.289 = 45.089 eSolutions Manual - Powered by Cognero Page 8 9-6 Applying Trigonometry The cliff is about 45 meters tall. 11. BALLOON The angle of depression from a hot air balloon to a person on the ground is 36°. When the person steps back 10 feet, the new angle of depression is 25°. If the person is 6 feet tall, how far above the ground is the hot air balloon to the nearest foot? SOLUTION: Let the horizontal distance from the hot air balloon to the person on the ground equal y and the height of the hot air balloon equal x. Since there are two unknowns, the sides opposite and adjacent, set up a tangent ratio for both situations and solve the system of equations. Set the two equations solved for x equal to each other and solve for y, the persons initial horizontal distance from the hot air balloon. Substitute the persons initial horizontal distance into the original equation to find the height of the balloon above the person. Now add the persons height to get the height of the balloon. 12.980 + 6 = 18.98 The hot air balloon is about 19 feet high. eSolutions Manual - Powered by Cognero Page 9 9-6 Applying Trigonometry 12. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40-foot ocean bluff near his home. He can see his two friends on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his friends are 34° and 48°, how far apart are his friends to the nearest foot? SOLUTION: Let the horizontal distance to the first friend equal x and the horizontal distance to the second friend equal y. Find x and y and then subtract to find the distance between friends. Now subtract the friends distances from the bluff. 68.198 – 41.419 = 26.779 The friends are about 27 feet apart. eSolutions Manual - Powered by Cognero Page 10 9-6 Applying Trigonometry Use trigonometry to find the area of △ABC to the nearest tenth. 13. ​ SOLUTION: The area of △ABC is about 62.3 units2. 14. ​ SOLUTION: The area of △ABC is about 41.8 units2. eSolutions Manual - Powered by Cognero Page 11 9-6 Applying Trigonometry 15. ​ SOLUTION: The area of △ABC is about 71.5 units2. 16. ​ SOLUTION: The area of △ABC is about 19.0 cm2. eSolutions Manual - Powered by Cognero Page 12 9-6 Applying Trigonometry 17. ​ SOLUTION: The area of △ABC is about 10.0 ft2. 18. ​ SOLUTION: The area of △ABC is about 35.6 yd2. eSolutions Manual - Powered by Cognero Page 13 9-6 Applying Trigonometry Use trigonometry to find the area of △ABC to the nearest tenth. 19. ​ SOLUTION: Because you do not know the measure of the included angle C, add the measures of angles A and B and subtract the total from 180. m∠C = 180 – (96 + 48) = 36° The area of △ABC is about 32.9 cm2. eSolutions Manual - Powered by Cognero Page 14 9-6 Applying Trigonometry 20. ​ SOLUTION: Because you do not know the measure of the included angle B, add the measures of angles A and C and subtract the total from 180. m∠B = 180 – (37 + 30) = 113° The area of △ABC is about 29.0 m2. eSolutions Manual - Powered by Cognero Page 15 9-6 Applying Trigonometry 21. ​ SOLUTION: Because you do not know the measure of the included angle A, add the measures of angles B and C and subtract the total from 180. m∠A = 180 – (75 + 53) = 52° The area of △ABC is about 106.4 ft2. eSolutions Manual - Powered by Cognero Page 16 9-6 Applying Trigonometry 22. USE ESTIMATION The angle of elevation to an airplane viewed from the air traffic control tower at an airport is 7°. The tower is 200 feet tall, and the pilot reports that the altitude of the airplane is 5127 feet. a. Explain how you can use angles of elevation and depression to estimate the distance from the air traffic control tower to the airplane. b. About how far away is the airplane from the air traffic control tower to the nearest foot? SOLUTION: a. Sample answer: Because the altitude of the plane and the height of the air traffic control tower are known, I can find the difference between these heights. Then, I can use this difference and the 7° angle of elevation to estimate the distance. b. The difference between the heights is 5127 – 200 = 4927 The airplane is about 40,429 feet away from the air traffic control tower. eSolutions Manual - Powered by Cognero Page 17 9-6 Applying Trigonometry 23. USE ESTIMATION A hiker dropped his backpack over one side of a canyon onto a ledge below. Because of the shape of the cliff, he could not see exactly where it landed. A park ranger is located on the other side of the canyon, at the same height, 113 feet away from the hiker. The ranger sights the backpack at an angle of depression of 32°. a. Explain how you can use angles of elevation and depression to estimate the distance the backpack fell. b. About how far down did the backpack fall to the nearest foot. SOLUTION: a. Because the horizontal distance between the hiker and the park ranger is known, I can use the 32° angle of depression to estimate the distance that the backpack fell. b. The backpack fell about 71 feet. 24. USE A MODEL Jermaine and John are standing 10 meters apart watching a helicopter hover above the ground. a. Find two different expressions that can be used to find h, the height of the helicopter. b. Equate the two expressions you found for part a to solve for x. Round your answer to the nearest hundredth. c. How high above the ground is the helicopter? Round your answer to the nearest hundredth. SOLUTION: a. For each right triangle, we can use the tangent ration since we are looking for the side opposite and know eSolutions Manual - Powered by Cognero Page 18 9-6 Applying Trigonometry something about the side adjacent. small right triangle: large right triangle: b. So, x = 34.98 m. c. Use the unrounded value of x from part b to find the height of the helicopter. The helicopter is hovering 49.95 meters above the ground. 25. USE A SOURCE Go online to research the Sandia Peak Tramway in New Mexico. If you were to stand at the top terminal of Sandia Peak Tramway and look at the base of the second tower along the tramway route, what would be the angle of depression for your line of sight? Round your answer to the nearest tenth of a degree. SOLUTION: eSolutions Manual - Powered by Cognero Page 19 9-6 Applying Trigonometry 26. REGULARITY A geologist wants to determine the height of a rock formation. She stands d meters from the formation and sights the top of the formation at an angle of x°, as shown. The geologist’s height is 1.8 meters. Write a general formula that the geologist can use to find the height h of the rock formation if she knows the values of d and x. SOLUTION: As shown in the picture, the height of the rock formation will be the height of the geologist plus the vertical distance from her line of sight to the top of the rock formation. The vertical distance is the side opposite and the horizontal distance is the side adjacent, so we must use the tangent ratio. Then add the geologists height: h = 1.8 + d tan x ° Find the area of △ABC to the nearest tenth. 27. ​ m∠A = 20°, c = 4 cm, b = 7 cm SOLUTION: The area of △ABC is about 4.8 cm2. eSolutions Manual - Powered by Cognero Page 20 9-6 Applying Trigonometry 28. ​ m∠C = 55°, a = 10 m, b = 15 m SOLUTION: The area of △ABC is about 61.4 m2. 29. a = 5.6 ft, c = 3.7 ft, m∠A = 37°, m∠C = 24° SOLUTION: Because you do not know the measure of the included angle B, add the measures of angles A and C and subtract the total from 180. m∠B = 180 – (37 + 24) = 119° The area of △ABC is about 9.1 ft2. eSolutions Manual - Powered by Cognero Page 21 9-6 Applying Trigonometry 30. a = 6.3 in., c = 7 in., m∠A = 42°, m∠C = 49° SOLUTION: Because you do not know the measure of the included angle B, add the measures of angles A and C and subtract the total from 180. m∠B = 180 – (42 + 49) = 89° The area of △ABC is about 22.0 in2. eSolutions Manual - Powered by Cognero Page 22 9-6 Applying Trigonometry 31. THEATER Albert is helping to build the set for a play. One piece of scenery is a large triangle that will be constructed out of wood and be painted to represent a mountain. Albert would like to know the area of the piece of scenery so that he can buy the right amount of paint. What is the area of this triangle? Round your answer to the nearest tenth of a square foot. SOLUTION: The area of the triangle is about 22.4 ft2. 32. FIND THE ERROR Terrence and Rodrigo are trying to determine the relationship between angles of elevation and depression. Terrence says that if you are looking up at someone with an angle of elevation of 35°, then they are looking down at you with an angle of depression of 55°, which is the complement of 35°. Rodrigo disagrees and says that the other person would be looking down at you with an angle of depression equal to your angle of elevation or 35°. Who is correct? Explain your reasoning. SOLUTION: Your horizontal line of site is parallel to the other person’s horizontal line of sight. So, the angles of elevation and depression are congruent according to the Alternate Interior Angles Theorem. Therefore, Rodrigo is correct. 33. CREATE A classmate finds the angle of elevation of an object, but she is trying to find the angle of depression. Write a question to help her solve the problem. SOLUTION: A question that can help your classmate solve the problem is "What is the relationship between the angle of elevation and the angle of depression?" eSolutions Manual - Powered by Cognero Page 23 9-6 Applying Trigonometry 34. ANALYZE Classify the statement below as true or false. Explain your reasoning. As a person moves closer to an object he or she is sighting, the angle of elevation increases. SOLUTION: True; sample answer: As a person moves closer to an object, the horizontal distance decreases, but the height of the object is constant. The tangent ratio will increase, and therefore the measure of the angle also increases. 35. PERSEVERE Find the value of x. Round to the nearest tenth. SOLUTION: Let z be the short leg of the triangles and y be the long leg of the smaller triangle. Set up two equations relating the opposite and adjacent sides of the triangles and solve both for z. Set the two equations solved for z equal to each other and solve for y. Add the two segments to find the length of the long leg of the larger triangle. 2 + 5.581 = 7.581 eSolutions Manual - Powered by Cognero Page 24 9-6 Applying Trigonometry Use the cosine ratio to find the hypotenuse of the larger triangle, x. So, the value of x is 7.8. 36. WRITE Describe a way that you can estimate the height of an object without using trigonometry. Explain your reasoning. SOLUTION: Suppose you sight something with a 45° angle of elevation. You do not have to use trigonometry to determine the height of the object. The legs of a 45°-45°-90° are congruent. So, the height of the object will be the same as your horizontal distance from the object. eSolutions Manual - Powered by Cognero Page 25