Expert: Scotto - 4/5/2009

Question
       You are on a fishing boat that leaves its pier and heads east. After traveling for 30 miles, there is a report warning of rough seas directly south.  The captain turns the boat and follows a bearing of S 45 degree W for 12 miles

a.  At this time, how far are you from the boat 's pier?

b. What bearing could the boat have originally taken to arrive at this spot?

Answer
a. The boat went 30 miles to the east.
It turns and heads south bearing 45 degrees west for 12 more miles.

So we have a triangle with one side 30 miles long
with an angle of 45 degrees, and then 12 miles almost back.

Since we know two sides of a triangle and the angle between,
what to do is call A 30, B 12, and the angle 45.

The equation is then the law of cosines,
which is C² = A² + B² -2AB*Cos(45).

So C² = 30² + 12² - 2*30*12*(1/√2).

This would mean that C were the √(900 + 144 - 720/√2).
Working all this out gives us √534.8831175,
which is roughly 23 miles.

To answer this one, I will do all the calculations in Excel
and just copy them over, but still leave them in Excel to do more.

b. To find the angle of this triangle from the start,
again we could use the law of cosines.

The two near sides would be A and C and the far side would be B.
This would gives us B² = A² + C² - 2ACcos(D),
where D is the angle we are looking for.

We know what B, A, and C are, so now to put them into the equation.

144 = 900 + 534.8831175 - 2*30*23.12754024*cos(D).
Now 2*30*23.12754024 = 1387.652414.  Lets add that number times
cos(D) to both sides and subtract off the 144 from both sides.

1387.652414*cos(D) = 900 + 534.8831175 - 144.
Now the right side is 1290.883118.

This will give cos(D) = 0.930264023
So the angle D is (and now I'll round) 21.5 degrees.