Two ships leave a harbor at the same time. One ship travels on a bearing Upper S 14 degrees Upper W at 15 miles per hour. The other ship travels on a bearing Upper N 75 degrees Upper E at 12 miles per hour. How far apart will the ships be after 3 hours?

69.94 miles

Step-by-step explanation:

The distance d₁ the first ship moves after 3 hours is 3 hours × 15 miles per hours = 45 miles

The distance d₂ the second ship moves after 3 hours is 3 hours × 12 miles per hours = 36 miles.

The angle the first ship's direction makes in the North-East direction is 90° - 75° = 15°

The angle the second ship's direction makes in the South-West direction  = 14°

The distance moved by the two ships form the side of a triangle. The angle,  θ between the two ship directions is 14° + 90° + 15° = 119°

Using the cosine rule, we find the distance d between the two ships

d = √(d₁² + d₂² -2d₁d₂cosθ)

= √(45² + 36² -2×45×36cos119°)

= √(2025 + 1296- (-1570.78))

= √(3321 + 1570.78)

= √4891.78

= 69.94 miles