Question

question
a boat leaves the marina and sails 13 miles northeast, then 12 miles east. in what direction must the boat sail to head directly back to the marina?
select the correct answer below:
66.55° w of s
23.45° w of s
45.00° e of n
66.55° e of n

Explanation:

Step1: Resolve northeast - direction vector

The northeast - direction (45° from the x - axis) vector of 13 miles has x - component $13\cos45^{\circ}=\frac{13}{\sqrt{2}}\approx9.19$ miles and y - component $13\sin45^{\circ}=\frac{13}{\sqrt{2}}\approx9.19$ miles. The second vector is 12 miles east (in the positive x - direction). So the total x - component of the displacement from the marina is $x = 9.19+12=21.19$ miles and the y - component is $y = 9.19$ miles.

Step2: Calculate the angle for the return vector

To return to the marina, we need to find the direction of the vector from the end - point to the starting - point. The angle $\theta$ of the return vector with the south direction (negative y - axis) can be found using the tangent function. $\tan\theta=\frac{21.19}{9.19}\approx2.306$. Then $\theta=\arctan(2.306)\approx66.55^{\circ}$ west of south.

Answer:

A. $66.55^{\circ}$ W of S