Question 1136159
<pre>
*  *
 *     *
  *         *       C
   *             *
 5  *                  *
     *                      *          
      *  123°                    *
       ******************************
                 7
</pre>

After 30 minutes (0.5 hours), the ship that has a speed of 10 mph has traveled 5 miles and the ship that has a speed of 14 mph has traveled 7 miles.  Given is that the two ships are 123° apart.  So, this all forms a triangle. (See above.)  We are looking for C.
<pr>
Use the law of cosines to find C:
c² = a² + b² - 2ab(cos c)
<pr>
Using our numbers:
c² = 5² + 7² - 2(5)(7)(cos 123°)
<pr>
c² = 25 + 49 - 70(cos 123°)
<pr>
c² = 74 - 70(cos 123°)
<pr>
c² = 74 - 70(-0.54464)
<pr>
c² = 74 + 38.1248
<pr>
c² = 112.1248
<pr>
c = 10.5889
<pr>
The two ships are 10.5889 miles apart after 30 minutes.