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Worksheet: Applications of the Law of Cosine

Q1:

The shown diagram represents a satellite calculating the distances and the angle. Find the distance between the two cities. Give your answer to one decimal place. Note that the diagram is not to scale.

Q2:

Two planes leave the same airport at the same time. One flies on a bearing of 2 0 at 500 miles per hour. The other flies on a bearing of 3 0 at 600 miles per hour. How far apart are the planes after 2 hours?

Q3:

A 113-foot tower is located on a hill that is inclined 3 4 to the horizontal, as shown in the figure. A guy wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed.

Q4:

A man ran 24 km down a straight road. He then turned 1 3 9 and ran a further 9 km along another straight road. Find, to the nearest kilometer, the shortest distance between his start and end points.

  • A 27 km
  • B 19 km
  • C 29 km
  • D 31 km

Q5:

An airplane flies 220 miles on a bearing of 4 0 and then changes course and flies 180 miles on a bearing of 1 7 0 . How far is the plane from its starting point and on what bearing? Give your answers to one decimal place.

  • A 52.9 miles, 9 2 . 9
  • B 172.9 miles, 5 2 . 9
  • C 6 1 4 1 . 9 miles, 3 7 . 1
  • D 172.9 miles, 9 2 . 9
  • E 362.9 miles, 1 0 2 . 5

Q6:

Two boats leave a harbor in different directions. One travels due west at 30 mph, and the other travels on a bearing of 2 9 5 at 42 mph. How far apart are they after 90 minutes?

Q7:

Two ships left a port at the same time. One traveled on a bearing of 3 2 0 at 18 mph, while the other traveled on a bearing of 1 9 4 at 22 mph. Find the distance between the ships after 10 hours. Give your answer to the nearest mile.

Q8:

A plane follows a straight path at a steady speed of 600 miles per hour. After 1 hour  30 minutes, the pilot makes a course correction of 1 0 , and the flight continues at the same speed, following a straight path, for another 2 hours. How far is the plane from its initial position? Give your answer to the nearest mile.