Question

1. the bearing of a lighthouse from a ship 9.2 kilometers away is 32°30. how far must the ship sail on a course 0° for the bearing of the lighthouse to be 122°30?

Explanation:

Step1: Identify the angle and side in the triangle

We have a triangle - like situation in navigation. The initial bearing from the ship to the lighthouse is $32^{\circ}30'$, and we want the bearing from the lighthouse to be $122^{\circ}30'$. The side - length of the initial distance of the ship from the lighthouse is $a = 9.2$ km. The angle between the initial direction of the ship and the new direction is $\theta=122^{\circ}30'-32^{\circ}30' = 90^{\circ}$.

Step2: Use the right - triangle relationship

Since the angle between the initial position of the ship and the new course is $90^{\circ}$, and the initial distance of the ship from the lighthouse is one side of a right - triangle, and the path of the ship on the new course is the hypotenuse of the right - triangle. In a right - triangle, if one side is $a = 9.2$ km and the angle between the sides is $90^{\circ}$, and we assume the distance the ship needs to sail is $d$.
We know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 90^{\circ}$, and the opposite side to the $90^{\circ}$ angle in our right - triangle situation is the distance the ship needs to sail.
Since the initial distance of the ship from the lighthouse is perpendicular to the new course, the distance $d$ the ship must sail is given by the Pythagorean theorem or simply, in this case, because of the right - angle, if we consider the initial distance as one side of a right - triangle, the distance $d$ the ship must sail is:
$d = 9.2$ km.

Answer:

$9.2$ km