PreCalc Bearing Word Problem (Need Sketch)

Question

An Airplane takes off was a bearing of 55ﾟ and flies 480 miles, after which it changes its course to a bearing of 285ﾟ. Finally the airplane flies back to its starting point with a bearing of 180ﾟ. Find the total mileage of flown by the airplane.

Answer 1

bearings are drawn from the north (vertical line)

draw a vertical line

draw a line going northeast at 55°

now you change course to a bearing of 285°

draw another vertical line at the endpoint of the first line segment

go 285° from the north and you are then moving northwest (up and to the left)

draw another vertical line

now you are moving vertically south at a bearing of 180° to get back to the starting point

you now have a triangle

call the northeast line "b" which is 480 miles

call the first vertical line "c"

call the northwest line "a"

use the law of sines

a/sin55°=480/sin75°=c/sin50°

angle 55° is given, angle C is 90-55=35, 35+15=50° (285 is 15 more than 270)

therefor angle B is 75°

so we have...

a/0.819=480/0.965=c/0.766

using the first two ratios...

a=(480)(0.819)/0.965

a=407.37 miles

using the second two ratios...

c=(480)(0.766)/(0.965)

c=381.01 miles

b=480 miles is given

total miles flown is 480+381.01+407.37=1268.38 miles

Answer 2

Draw a picture....To best visualize the problem, make it at least roughly to scale

On a standard X-Y graph, a bearing of 0 is the +X direction. +Y is a bearing of 90 degrees, -X is 180 degrees, etc.

The plane starts moving up and to the right, and finishes moving the the left. The second leg is down and slightly to the right. (a bearing of 270 degrees would be straight down.

Since we are given all the angles, all that is needed is to apply the law of sines to get the length of the 2 unknown legs. (first, use the bearings to get the 3 internal angles of the triangle)