Law of Sines and Cosines Applications Answers

Honors Pre-Calculus Law of Sines & Cosines Applications Name Where appropriate, give angle measures to the nearest tenth of a degree and lengths of sides in simplest radical form or to the nearest hundredth. 1. Find the area of if Also find . 1. Area=______________ ________________ 2. 3. Find the measure of the largest angle in a triangle with sides having lengths √ √ and √ . 2. ___________________ In 3. ________________ , Find . ________________ 4. Three measurements in are given as Show that at least one of the measurements is incorrect. 5. A regular polygon with sides is inscribed in a circle with radius . Find its area. Compare your answer with . 6. 7. 5. ___________________ A submarine dives at an angle of with the horizontal. If it takes minutes to dive from the surface to a depth of feet, how fast does it move along its sloping path downward? Give your answer in feet per minute. Then convert it to nautical miles per hour. Note: nautical mile per hour feet per hour). 6. ___________________ In , Find the measure(s) of . 7. ______________ 8. In parallelogram 9. , . a. Find the area of . b. Find the lengths of both diagonals. A triangle has an area of and two of its sides are and Find the possible measures of the angle formed by these sides. 10. In the diagram given below, is similar to a. Find the lengths b. Find the ratio of the areas of the triangles. C a and 8a. ___________________ 8b. ___________________ 9. ___________________ long. . 10. . _______________ ________________ ________________ ratio = _____________ F 6 e A 11. B D f E The diagonals of a parallelogram have lengths and and they meet at a angle. Find the area and perimeter of the parallelogram. 11. Area =______________ Perimeter=__________ 12. An obtuse triangle with area has two sides of lengths and Find the length of the third side. There are two answers. 12. ___________________ 13. The perimeter of a regular decagon ( . Find its area. 13. ___________________ 14. If fencing costs per foot, how much will it cost to buy fencing to go around the plot of land shown below? 14. ___________________ ft ft sides) is . 15. 16. 17. In the township of Madison, rural undeveloped land is taxed at a rate of per acre. Find the tax on the plot of land shown. Note: acre ft2 15. ___________________ A ship is steaming north at knots ( nautical miles per hour) when the captain sights a small island at an angle of to the east of the ship’s course. After minutes, the angle is . How far away is the island at this moment (in nm)? 16. ___________________ In , a. Solve . . 17a. ______________ ______________ ______________ 18. b. Find the area of . c. Find the length of the altitude to ̅̅̅̅ . Ship sights ship on a compass bearing of compass bearing of ship from ship . 19. Ship sights ship on a bearing of 20. An airplane flies on a course of at a speed of east of its starting point is it after 2 hours? 17b. Area = _____________ 17c. ___________________ 18. ___________________ 19. ___________________ 20. ___________________ . Make a sketch and give the . What is the bearing of from ? km/h. How far 21. 22. A hunter walks east for hour and then north for hours. What course should the hunter take to return to his starting point? What assumptions do you make to answer the question? 21. ___________________ Point is km north of point , and point is km north of point a bearing of from . Find the bearing and distance of from . 22. Bearing_____________ on Distance____________ 23. Point is km west of point , and point Find the bearing and distance of from is km southwest of . 23. Bearing_____________ Distance____________ 24. 25. 26. Traveling at a speed of knots, a ship proceeds south from its port for hours and then changes course to for hours. At this time, how far from port is the ship? 24. ___________________ A sailboat leaves its dock and proceeds east for miles. It then changes course to until it is due south of its dock. How far south is this? 25. ___________________ Two ships, and , leave port at the same time. Ship proceeds at knots on a course of , while ship proceeds at knots on a course of . After hours, ship loses power and radios for help. How far and on what course must ship travel to reach ship . 26. Bearing_____________ Distance____________ Sketch each plot of land described, and find its AREA. 27. 28. 29. From an iron post, proceed meters northeast to the brook, then meters east along the brook to the old mill, then meters to a post on the edge of Wiggin’s Road, and finally along Wiggin’s Road back to the iron post. 27. Area=______________ From a granite post, proceed ft east along Tasker Hill Road, then along a bearing of for ft, then along a bearing of for ft, and finally along a line back to the granite post. 28. Area=______________ From a cement marker, proceed m southwest to the river, then m south along the river to the bridge, then m to a sign on the edge of Sycamore Lane, and finally along Sycamore Lane back to the cement marker. 29. Area=______________ Answers: Law of Sines and Cosines Applications 1. Area = ; 2. 4. Sine cannot be greater than 5. 7. 8a. A = 9. 10a. 11. A = sq. units P= units 14. 16. nm 29. 6. nmh 8b. and 13. sq. units sq. ft.; Tax = 17a. sq. units or 23. Bearing: , 10b. 12. 17c. √ 19. 21. Assumption: Rate is constant; Bearing: 26. sq. units 15. A = 17b. 18. sq. units 3. ; TR = or or 24. 27. 20. km 22. nm 25. 28. km miles HPC 1. Law of Sines and Cosines Review for Quiz 6.1-6.2/Packet Name______________________ An airplane flies on a course of 130 at a speed of 1100 km/h. How far east of its starting point is it after 3 hours? 1.)___________________ 2. One angle of an isosceles triangle has a measure of 150. If the area of the triangle is 9 cm2, what is the perimeter of the triangle? 2.)___________________ 3. A ship leaves port and proceeds west 30 miles. It then changes course to 020 until it is due north of its origin. How far north of its origin is it? 3.)___________________ 4. The area of ∆ABC is 45 square units. If a = 10 and b = 15, find the measure of angle C to the nearest degree. 4.)___________________ 5. Given the diagram below, find ZN to the nearest whole unit. 5.)___________________ Y 2 N 5 4 Z 6. 7 X Solve XYZ if x = 52, y = 70 and z = 100. 6.)___________________ _____________________ _____________________ 7. Solve XYZ and find its area if  X  52,  Y  70 , and z  100 . 7.)___________________ _____________________ _____________________ 8. Solve XYZ and find its area if  X  52, y  70, , and x  100 . 8.)___________________ _____________________ _____________________ 9. 10. 11. Eli Cooley flew his plane 900 km north, turned 15 , and flew 1150 km. How far is Mr. Cooley from his starting point? 9.)___________________ The angle of elevation to the peak of a mountain is 30 . A kilometer closer, the angle of elevation is 35 . Find the height of the mountain. 10.)__________________ A lakefront plot of land is shown below. What is its area and lakefront footage (bolded sides are lakefront footage)? 11.)__________________ 70 80 250 ft 350 ft 75 12. The diagram shows the dimensions for a sail on a wooden model ship. Find the area of the sail to the nearest square inch. [The angle at the bottom left is NOT a right angle!] 12.)__________________ 12.5 in 26 in 18 in. 75 16.5 in 13. 14. 15. A ship leaves port and travels 36 miles west, then 24 miles on a course bearing 213 . How far is it from its starting point? 13.)__________________ A hiker walks 8000 m on a course of S 81 E . She then changes direction and hikes 5000 m on a course of N 32W . How far is she from her starting point, and on what course must she travel to return to the starting point? 14.)__________________ A regular pentagon is inscribed in a circle of radius 4 in. Find the area of the pentagon. 15.)__________________ 16. Observers at points A and B, 30 km apart, sight an airplane between them at angles of elevation of 40 and 75, respectively. How far is the plane from each observer? 16.)__________________ 17. Two hikers follow a trail that splits into two forks. Each hiker takes a different fork. The forks diverge at an angle of 67 and both hikers walk at a speed of 3.5 mph. How far apart are the hikers after 1 hour? 17.)__________________ 18. After leaving an airport, a plane flies for 1.5 hours at a speed of 200 km/h on a course of 200. Then, on a course of 340, the plane flies for 2 hours at a speed of 250 km/h. At this time, how far from the airport is the plane? 18.)__________________ 6.  X  29.4,  Y  41.4,  Z  109.2 7. x  92.9, y  110.8,  Z  58, AREA  4364.9 8.  Y  33.5,  Z  94.5, z  126.5, AREA  3489.2 9. 2032.7 km 2 2 10. 3.3 km 11. Area = 55,132.21 ft , lakefront footage = 401.3 ft 12. 301.8 in 13. 53.04 miles 2 14. 6042.8 m, bearing of 240 or 15. 38.04 in 16. 31.97 km from A and 21.3 km from B 17. 3.86 miles 18. 331.9 km 1. 2527.9 km 2. 23.6 cm 3. 82.4 miles 4. 36.9 or 143.1 5. 5.0