Question

watch the video and then solve the problem given below. click here to watch the video. a boat leaves the entrance to a harbor and travels 125 miles on a bearing of n 50° e. how many miles north and how many miles east from the harbor has the boat traveled? the boat has traveled 80.3 miles north. (round to the nearest tenth of a mile.) the boat has traveled miles east. (round to the nearest tenth of a mile.)

Explanation:

Step1: Define trigonometric relationships

We can use right - triangle trigonometry. If the distance traveled by the boat is the hypotenuse $d = 125$ miles and the bearing is N $50^{\circ}$E. The north - south distance $y$ and east - west distance $x$ form a right - triangle with the path of the boat. The angle $\theta=50^{\circ}$, and we know that $\sin\theta=\frac{x}{d}$ and $\cos\theta=\frac{y}{d}$.

Step2: Calculate the east - ward distance

We want to find the east - ward distance $x$. Since $\sin\theta=\frac{x}{d}$, and $d = 125$ miles, $\theta = 50^{\circ}$, then $x=d\sin\theta$. Substituting the values, we have $x = 125\times\sin(50^{\circ})$.
Using a calculator, $\sin(50^{\circ})\approx0.766$, so $x=125\times0.766 = 95.75\approx95.8$ miles.

Answer:

95.8