* *
* *
* * C
* *
5 * *
* *
* 123° *
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7
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.
Use the law of cosines to find C:
c² = a² + b² - 2ab(cos c)
Using our numbers:
c² = 5² + 7² - 2(5)(7)(cos 123°)
c² = 25 + 49 - 70(cos 123°)
c² = 74 - 70(cos 123°)
c² = 74 - 70(-0.54464)
c² = 74 + 38.1248
c² = 112.1248
c = 10.5889
The two ships are 10.5889 miles apart after 30 minutes.