## anonymous 5 years ago if a two ships left the same port at the same time but ship A left at N35'E45 degrees at a speed of 25 mph and ship B left at N24'56degrees at a speed of 25 mph, how far apart would they be in 4 hours? i dont understand what formula to use law of sines? law of cosines? help!!

1. anonymous

Hi, this site keeps crashing on me. Use cosine rule - the ships will have an angle between them equal to the difference of their angles, and they will have traveled 100miles in 4 hours, so you have your side lengths. The distance is the distance between them .

2. anonymous

the second one is actually S24'E56 so so do i find the hypotenuse?

3. anonymous

Cosine rule...

4. anonymous

If you can wait, I can send you a solution later. I can't do anything on the machine I'm on right now.

5. anonymous

yes i can wait! and ship b is 23 mph sorry for all of the mistakes

6. anonymous

it's okay. later.

7. anonymous

Okay, this is going to be an application of the cosine rule. Do you know how to get angles when they're given in this form first?

8. anonymous

yes, you put them in the calculator using 2nd angle

9. anonymous

Yeah, but I mean geometrically. I'm drawing a picture of the situation.

10. anonymous

no and ok awesome

11. anonymous

12. anonymous

Now, the angle (when you're given it in this form, i.e. North blah blah) is swept out from the north pole towards where the object is. Here, ship B has the smaller angle (blue) and ship A has the larger. What you're being asked to find is the distance between the ships at the end of 4 hours.

13. anonymous

The distance is the line in pink. Now, since speed is *defined* as $(speed)=\frac{(distance)}{(time)}$it means,$(distance)=(speed) \times (time)$

14. anonymous

oh, but wouldnt ship b be in the 4th quadrant since its angle is SE?

15. anonymous

oh ok, so then you would get 100 mi and 92 mi which give you the side lengths?

16. anonymous

The distance ship A travels in 4 hours is then,$d_A=25\frac{mi}{h}\times 4 h=100 miles$

17. anonymous

Ship B will, in 4 hours, travel,$d_B=23\frac{mi}{h} \times 4 h = 92 miles$

18. anonymous

Now you have the two distances - the blue and red lines.

19. anonymous

All you need to do is now find the angle BETWEEN the two ships - this will stay the same since they don't change course. The angle between them is the angle in pink. It's given by the difference between the larger and smaller angles.

20. anonymous

$\theta = 35^o45' -24^o56'=10^o49'$

21. anonymous

You fine so far?

22. anonymous

ok that makes sense, but what if the angle was supposed to be in the 4th quadrant since i wrote it wrong

23. anonymous

i mean the 3rd

24. anonymous

I'm not sure what you mean.

25. anonymous

since ship b was S24E56 but i wrote it as N24E56

26. anonymous

like ship b was supposed to be S24E56 not N24E56

27. anonymous

Give me a second.

28. anonymous

ok

29. anonymous

30. anonymous

To say S24E56 means "24 degrees, 56 minutes from magnetic South in the Easterly direction." So you move from the 'South' line and sweep upwards towards the East by 24 degrees, 56 minutes.

31. anonymous

The first letter tellls you what pole to start from, then the second letter tells you whether to sweep left or right (left = West, right = East).

32. anonymous

The problem is set up in the same way - you're still going to have to use the cosine rule because you need to find the distance opposite an angle, where that angle is BETWEEN two sides you know the value of.

33. anonymous

so when im finding the angle, do i have to include the 56 minutes part when using the law of cosine?

34. anonymous

35. anonymous

Is this what you've got?

36. anonymous

yes, exactly!

37. anonymous

Okay...

38. anonymous

Now we have to find the angle IN BETWEEN from the info. we're given.

39. anonymous

so to do this, do i subtract the two give angles from 180?

40. anonymous

41. anonymous

Note, from your picture, that the red angle and the green angle will sum to 90 degrees. The blue angle and purple angle will sum to 90 degrees also.

42. anonymous

We need (green) + (purple), since this is the angle in between. We have, (red) + (green) + (purple) + (blue) = 90 + 90 = 180 degrees. We know that (red) = 35degrees, 45 minutes and that (blue) = 24 degrees, 56 minutes, so

43. anonymous

(green) + (purple) = 180 - 35deg.45min - 24deg.56min. = 119deg.19min.

44. anonymous

This is your angle in between the two lines. We have the distances calculated above, so we have the following now:

45. anonymous

46. anonymous

(They're both the same). Okay, so now we can use the cosine formula: $a^2=100^2+92^2-2(100)(92)\cos(119^o19')$to get$a^2 \approx 27473$which means that the distance is (after taking the square root),$a \approx 165.75$miles.

47. anonymous

That's it.

48. anonymous

thank you. i totally understand it now!

49. anonymous

Good :) Good luck with it. If you need anymore help, come back to the site - I might be around.

50. anonymous

ok. thank you!! =)