Answer to part a): 76.4068348682472 miles
Answer to part b): 177 degrees
Note: The answer for part a) is approximate. Round this value to whatever you need to.
-------------------------------------------------------------------------------------------------
-------------------------------------------------------------------------------------------------
Explanation:
First set up the drawing
Step 1) Plot the harbor (point H). Then draw a straight line pointing directly north to point N. This will represent true north on the compass. The bearing angles will originate from this direction
Step 2) Face true north and rotate 15 degrees clockwise as shown by the red angle. Draw a segment that goes from point H to point A. Label this segment 32 (to represent ship A has traveled 32 miles). So AH = 32. Angle NHA = 15 degrees.
Step 3) Turn back to face true north again. Draw a segment HB such that angle NHB is 165 degrees as shown by the blue angle. Denote HB = 47 to indicate that ship B has traveled 47 miles.
Step 4) Draw a segment from A to B. The goal is to find the length of this segment. It's unknown for now, so we'll call it x.
After those steps are completed, you should get something like this
The summary information of that drawing is
point H = harbor
point A = ship A's location (after traveling 32 miles from the harbor)
point B = ship B's location (after traveling 47 miles from the harbor)
point N = true north
--------
segment HA = 32 miles
segment HB = 47 miles
segment AB = x miles
--------
angle NHA = 15 degrees (red angle)
angle NHB = 165 degrees (blue angle)
----------------------------------------------------------------------
We can use the law of cosines to find the value of x. But first we need to figure out what angle AHB is equal to. The angle addition postulate says
(angle AHB) + (angle NHA) = (angle NHB)
Plug in the known angles and isolate "angle AHB"
(angle AHB) + (angle NHA) = (angle NHB)
(angle AHB) + (15) = (165)
(angle AHB) + 15 = 165
(angle AHB) + 15-15 = 165-15
angle AHB = 150
Let's update the drawing. I'm going to erase the other two angles (15 and 165 degrees) since they won't be needed further. In their place, I'll add in angle AHB = 150 degrees. I'll also erase point N and the arrow pointing north since we don't need that anymore either.
We now have this
The angle C = 150 is between the two sides of a = 32 and b = 47
Using the law of cosines, we can say
c^2 = a^2 + b^2 - 2*a*b*cos(C)
x^2 = 32^2 + 47^2 - 2*32*47*cos(150)
x^2 = 5838.0044145836
sqrt(x^2) = sqrt(5838.0044145836)
x = 76.4068348682472
Therefore the distance from A to B is approximately 76.4068348682472 miles. Round this value to whatever you need to or however the teacher requires. This is the answer to part A
---------------------------------------------
To get the bearing from the first ship to the second ship, we need to add on a bit to the drawing. Let's draw a segment that goes straight up from point A. Plot C on this segment.
Also, extend out segment HA (as shown by the dashed black line) and plot D on this dashed segment.
I'll mark the bearing as the angle y. It's unknown for now.
Angle z = angle DAB
AC || NH, so this means that
angle CAD = angle NHA = 15 degrees
Use the law of cosines to find angle A
sin(A)/a = sin(H)/h
sin(A)/47 = sin(150)/x
sin(A)/47 = sin(150)/76.4068348682472
sin(A) = 47*sin(150)/76.4068348682472
sin(A) = 0.30756410785138
A = arcsin(0.30756410785138)
A = 17.912493477208
So this means
angle NAB = 17.912493477208 degrees
angle DAB = 180 - (angle NAB)
angle DAB = 180 - (17.912493477208)
angle DAB = 162.087506522792
angle CAB = (angle CAD) + (angle DAB)
angle CAB = (15) + (162.087506522792)
angle CAB = 177.087506522792
which rounds to 177 degrees
so y = 177 roughly