Draw a horizontal line, of length 8, representing the heading of the ship. Draw a second line, no set length, at 30 degrees above the first line ("due east" is 90 degrees from North so 60 degrees is 90- 60= 30 degrees from due east), representing the true path. Using compasses, strike a circle with center at the tip of the horizontal line of radius 4, representing the current of unknown direction. That circle will strike the line at 30 degrees

1) not at all, if the radius is not large enough

2) exactly once if the radius is exactly enough to just touch the line

3) twice if the radius is greater than that distance (this is the "ambiguous case".)

The "ground speed" of the ship is the length of the third side of that triangle.

Use the "cosine law". It says that if a triangle has sides of length a, b, and c with angles A, B, and C, each opposite the corresponding lower letter, then

.

Since the only angle we know is the 30 degree angle, lets use that as "C". Then c= 4, a= 8 and b is unknown:

A quadratic equation may have 0, 1, or 2 real roots corresponding to the three geometric cases.

Go through that and then try the second problem yourself.