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anonymous

  • one year ago

Suppose you are standing on the edge of a canyon that is 400 meters high, where two sides of the canyon are at the same height and the walls of the canyon are perfectly vertical. The angle of depression to the bottom of the other side of the canyon is 14.3 degrees. Find the distance to the other side of the canyon. Ignore the height in the computations.

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  1. anonymous
    • one year ago
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    basic tip: the angle of depression is the angle from the horizontal looking down at an object

  2. mathstudent55
    • one year ago
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    |dw:1437107516201:dw|

  3. anonymous
    • one year ago
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    why are they right angles

  4. mathstudent55
    • one year ago
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    |dw:1437107592620:dw|

  5. mathstudent55
    • one year ago
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    Because the walls of the canyon are perfectly parallel and the same height.

  6. anonymous
    • one year ago
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    ohhhhhh ok

  7. mathstudent55
    • one year ago
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    Since the walls are perfectly vertical, they are both perfectly perpendicular to the ground. The ground is horizontal. Two lines perpendicular to the same line are parallel to each other.

  8. mathstudent55
    • one year ago
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    Ok.You understand the drawing?

  9. anonymous
    • one year ago
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    i get it now, i was confused on that part

  10. mathstudent55
    • one year ago
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    Ok. It means we have a rectangle. We are looking for x. |dw:1437107821415:dw|

  11. mathstudent55
    • one year ago
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    We can use trig of right triangles to find x.

  12. mathstudent55
    • one year ago
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    |dw:1437107892862:dw|

  13. mathstudent55
    • one year ago
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    The horizontal line at the top of the canyon, and the horizontal line at the floor of the canyon are parallel lines. We already know this because they are opposite sides of a rectangle. The line of sight is a transversal. This way we know the bottom right angle (marked above) measures 14.3 deg because the two 14.3 deg angles are alternate interior angles of parallel lines cut by a transversal.

  14. mathstudent55
    • one year ago
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    Now we have a simple right triangle trig problem.

  15. anonymous
    • one year ago
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    ok

  16. mathstudent55
    • one year ago
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    We are looking for x. |dw:1437108150647:dw|

  17. mathstudent55
    • one year ago
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    Do you remember the sine, cosine, and tangent ratios?

  18. anonymous
    • one year ago
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    yeah

  19. mathstudent55
    • one year ago
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    Do you use SOHCAHTOA to remember it?

  20. anonymous
    • one year ago
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    yes

  21. mathstudent55
    • one year ago
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    Great. Here we have the 14.3 angle on the bottom right. For the 14.3 angle, is the x side the adjacent or the opposite leg?

  22. anonymous
    • one year ago
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    adjacent

  23. anonymous
    • one year ago
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    and 400 is the opposite

  24. anonymous
    • one year ago
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    so we use tangent

  25. mathstudent55
    • one year ago
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    Great. That makes the 400 m side the opposite leg since the hypotenuse is unlabeled and we are not interested in it.

  26. mathstudent55
    • one year ago
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    Yes, we use the tangent.

  27. anonymous
    • one year ago
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    so basically I got 101.958

  28. anonymous
    • one year ago
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    so 102f

  29. anonymous
    • one year ago
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    102m

  30. mathstudent55
    • one year ago
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    I don't get that.

  31. anonymous
    • one year ago
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    what did you get

  32. mathstudent55
    • one year ago
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    Let me show you how just by logic and thinking you can tell that answer cannot be correct.

  33. mathstudent55
    • one year ago
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    You remember the good old 45-45-90 triangle?

  34. anonymous
    • one year ago
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    OHHHHHHHH

  35. anonymous
    • one year ago
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    RIGHT

  36. anonymous
    • one year ago
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    |dw:1437108626021:dw|

  37. mathstudent55
    • one year ago
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    If we had a 45-45-90 triangle, the legs would be congruent, so x would also be 400 m, right? |dw:1437108626987:dw|

  38. mathstudent55
    • one year ago
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    |dw:1437108699037:dw|

  39. anonymous
    • one year ago
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    so basically the distance to teh other side is 400 meters

  40. mathstudent55
    • one year ago
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    No. I'm just explaining why 102 m cannot be correct. The answer has to be a number much larger than 400 m.

  41. mathstudent55
    • one year ago
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    In the 45-45-90 case, the two legs would be 400 m. We don't have a 45-45-90 case.

  42. anonymous
    • one year ago
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    ok

  43. mathstudent55
    • one year ago
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    Since our bottom right angle is only 14.3 deg, the upper left angle is its complelement, so it's 75.7 deg.

  44. anonymous
    • one year ago
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    ok

  45. anonymous
    • one year ago
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    if possible could you draw your steps

  46. mathstudent55
    • one year ago
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    |dw:1437108865822:dw|

  47. mathstudent55
    • one year ago
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    Wit a 45-45-90 triangle, x is the same as 400 m. Now see how our case more or less looks like above. As the top left angle gets larger and larger than 45 deg, the distance x also increases.

  48. mathstudent55
    • one year ago
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    Since 75.7 is much larger than 45, x is much larger than 400 m.

  49. anonymous
    • one year ago
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    ok

  50. mathstudent55
    • one year ago
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    Your last equation is incorrect bec you have adj/opp. The tan ratio is opp/adj. \(\tan \theta = \dfrac{opp}{adj} \) \(\tan 14.3^o = \dfrac{400~m}{x} \) \(x = \dfrac{400~m}{\tan 14.3^o}\)

  51. mathstudent55
    • one year ago
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    All you need to do is calculate x.

  52. anonymous
    • one year ago
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    1569.3

  53. mathstudent55
    • one year ago
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    You got it!

  54. mathstudent55
    • one year ago
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    Don't forget the units.

  55. anonymous
    • one year ago
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    THNAK YOU SO MUCH

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