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 miles north. (round to the nearest tenth of a mile.)

Explanation:

Step1: Define the components

We can use trigonometry. Let the distance traveled be the hypotenuse of a right - triangle. The angle with the north direction is $\theta = 50^{\circ}$, and the hypotenuse $d = 125$ miles. The north - south displacement is given by $y$ and the east - west displacement by $x$.

Step2: Calculate the northward distance

The formula for the northward distance (adjacent side to the angle $\theta$ with respect to the hypotenuse) is $y=d\cos\theta$. Substituting $d = 125$ miles and $\theta = 50^{\circ}$, we have $y = 125\times\cos(50^{\circ})$.
$y=125\times0.6427876097$
$y\approx80.3$ miles

Step3: Calculate the eastward distance

The formula for the eastward distance (opposite side to the angle $\theta$ with respect to the hypotenuse) is $x = d\sin\theta$. Substituting $d = 125$ miles and $\theta=50^{\circ}$, we get $x = 125\times\sin(50^{\circ})$.
$x=125\times0.7660444431$
$x\approx95.8$ miles

Answer:

The boat has traveled approximately 80.3 miles north.