Question

A ship is sailing due north. At a certain​ point, the bearing of a lighthouse 3.7 km away is N37.6degreesE. Later​ on, the captain notices that the bearing of the lighthouse has become Upper S 34.5 degrees E.

How far did the ship travel between the two observations of the​ lighthouse?

Answer 1

The ship traveled approximately 5.91 kilometers between the two observations of the lighthouse.

From the information, we can identify the following angles:

Angle BAC = 37.6° (initial bearing)

Angle BCA = 34.5° (final bearing)

Find the right triangle

Since the ship is sailing due north, the two legs of the right triangle formed by the lighthouse, the initial position of the ship, and the final position of the ship are perpendicular to each other. Therefore, we can use the tangent function to relate the angles and the sides of the triangle.

Calculate the distance traveled

The distance traveled by the ship is equal to the length of the hypotenuse of the right triangle. Using the tangent function, we can calculate the hypotenuse as follows:

tan(37.6°) = (AB) / (BC)

or

BC = AB * tan(37.6°)

Plugging in the known value, we get:

BC = 3.7 km * tan(37.6°) ≈ 4.69 km

However, this is only the length of one leg of the right triangle. To find the total distance traveled, we need to use the Pythagorean theorem.

AC² = AB² + BC²

or

AC = √(AB² + BC²)

Plugging in the known values, we get:

AC = √(3.7 km² + 4.69 km²) ≈ 5.91 km

Answer 2

6.2 km

Explanation:

Let x be the distance travel by ship between the two observations of the lighthouse.

AC=3.7 km

`\angle B=34.5^(\circ)`

`\angle A=37.6^(\circ)`

In triangle ABC,

`\angle A+\angle B+\angle C=180^(\circ)`

Triangle angle sum property

Substitute the values

`37.6+34.5+\angle C=180`

`\angle C=107.9^(\circ)`

sin law:
`(a)/(sin A)=(b)/(sin B)=(c)/(sin C)`

Taking
`(b)/(sin B)=(c)/(sin C)`

Substitute the values then we get

`(3.7)/(sin 34.5)=(x)/(sin 107.9)`

`x=(3.7* sin 107.9)/(sin 34.5)`

`x=6.2 km`

Hence, the ship travel between the two observations of the light house=6. 2 km