Hello, fayeorwhatsoever!

3) A ship leaves a port and sails for 3 hrs on a course N78°E at 12 knots.

Then the ship changes its course to 168° and sails for 5 hours at 10 knots.

After 8 hours, how far is the ship from the port?

Determine the bearing of the ship from the port. Code:

N M
: :
: :
: A o 168°
: * *
: * 90° *
: * *
:78°* 36 * 50
: * *
: * *
:* *
P o *
* *
* *
* *
* *
* *
o B

The ship starts at port $\displaystyle P$. .$\displaystyle \angle NPA = 78^o$

It sails 36 nautical miles to $\displaystyle A$.

It changes course to $\displaystyle \angle MAB = 168^o$

. . and sails 50 nautical miles to $\displaystyle B$.

Note that $\displaystyle \angle PAB = 90^o.$

Pythagorus: .$\displaystyle PB^2 \:=\:36^2 + 50^2 \:=\:3796 \quad\Rightarrow\quad PB \:=\:\sqrt{3796} \:\approx\:61.6$

The ship is about 61.6 nautical miles from the port.

In right triangle $\displaystyle PAB\!:\;\;\tan(P) \:=\:\tfrac{50}{36} \quad\Rightarrow\quad P \:=\:\tan^{-1}\left(\tfrac{50}{36}\right) \:\approx\:54.2^o $

The bearing is: .$\displaystyle \angle NPB \;=\;78^o + 54.2^o \;=\;132.2^o$