Question

A goose flying south for the winter travels 55 miles directly south east (45 degrees) and then flies at an angle of 35 degrees Northeast for 75 miles. How far and in what direction did the goose travel?

Answer 1

Part A

The distance from the start point the goose travels is approximately 84.96 miles

Part B

The direction the goose travel is approximately 71° Northeast

Step-by-step explanation:

Part A

The distances the goose flies is given as follows;

The distance directly 45 degrees southeast the goose travels, AB = 55 miles

The distance Northeast the goose travels, BC = 75 miles

The angle Northeast the goose travels = 35°

From the attached drawing of the path the goose travels, we have, by cosine rule, the following relation;

a² = b² + c² - 2·b·c·cos(A)

Therefore, we have;

The length of side
`\overline{AC}`² = 55² + 75² - 2 × 55 × 75 × cos(80°) = 7217.40253425

`\overline{AC}` = √(7217.40253425) ≈ 84.9552972701

The distance from the start point the goose travels ≈ 84.96 miles

Part B

By sine rule, we have;

75/(sin(A)) =
`\overline{AC}`/(sin(80°))

sin(A) = (75/
`\overline{AC}`) × (sin(80°)

∴ sin(A) = (75/84.9552972701) × (sin(80°) = 0.8694052016

∠A = arcsin(0.8694052016) = 60.3895991264° ≈ 60.4°

The direction the goose travel = 180 - 64 - 45 = 71°

∴ The direction the goose travel =≈ 71° Northeast