A diving company is looking for a shipwreck 35 m below the surface of the ocean. The command ship locates the wreck at 55 degrees South of West at an angle of depression of 18 degrees reference the surface. The command ship notifies the diver, who is located 27 degrees South of East at an angle of depression of 30 degrees. How far must the diver travel to arrive at the wreck?

The distance between the diver and the shipwreck is approximately 130.76 m

Explanation:

The depth of the shipwreck below the surface of the water = 35 m

The location of the wreck = 55° South of West

The angle of depression of the wreck = 18°

The location of the diver = 27° South of East

The angle of depression of the diver = 30°

The distance of the wreck from the ship = 35 m/(sin(18°)) ≈ 113.26 m

The horizontal distance of the wreck from the ship = 35 m/(sin(18°)) × (cos(18°)) ≈ 107.72 m

The x-coordinate of the shipwreck, x₁ = 107.72 m × cos(55°) ≈ 61.79

The y-coordinate of the shipwreck, y₁ = 107.72 m × sin(55°) ≈ -88.24

The z-coordinate of the shipwreck, z₁ = -35

The distance of the diver from the ship = 35 m/(sin(30°)) = 70 m

The horizontal distance of the diver from the ship = 70 m × cos(30°) ≈ 60.62 n

The x-coordinate of the diver from the ship, x₂ = -60.62 m × cos(27°) ≈ -54.013

The y-coordinate of the diver from the ship, y₂ = 60.62 × sin(27°) ≈ -27.52

The z-coordinate of the diver from the ship, z₂ = -35

The distance between the diver and the shipwreck, 'd', is given by the distance between two points given the x, y, z coordinates as follows;

d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

Therefore, we have;

d = √((61.79 - (-54.013))² + (-88.24 - (-27.52))² + (35 - 35)²) ≈ 130.76

The distance between the diver and the shipwreck, d ≈ 130.76 m.