Question

A boat leaves the entrance to a harbor and travels 175 miles on a bearing of N 52° E. How many miles north and how many miles east from the harbor has the boat traveled?

Answer 1

The boat has traveled 107.7 miles north

The boat has traveled 137.9 miles east

Step-by-step explanation:

If the boat travels 175 miles on a bearing of N 52° E, we can represent the situation as follows

So, it forms a right triangle and we can use the trigonometric functions to know how much it travels north and east.

Since the north distance is the adjacent side of the given angle, we get

`\begin{gathered} \text{cos}52=\frac{\text{Adjacent}}{Hypotenuse} \\ \cos 52=(y)/(175) \\ 175\cdot\cos 52=y \\ 175(0.62)=y \\ 107.7\text{ mil}es\text{ = y} \end{gathered}`

Then, the boat has traveled 107.7 miles north

In the same way, the distance to the east is the opposite side, so

`\begin{gathered} \sin 52=(Opposite)/(Hypotenuse) \\ \sin 52=(x)/(175) \\ 175\cdot\sin 52=x \\ 175(0.79_{})=x \\ 137.9\text{ miles = x} \end{gathered}`

So, the boat has traveled 137.9 miles east