Geometry

A surveyor sights the far bank of a river at an angle of 110˚ to the near bank. She then moves 75 feet upriver and sights the same point on the far bank of the river at an angle of 150˚. What is the shortest distance across the river?

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  1. Hmmm... Strange labeling. Still, using the acute angles, I get that the distance d can be found using

    d cot30˚ - d cot70˚ = 75
    so,
    d = 75/(cot30˚ - cot70˚)

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  2. On my sketch I assumed the two sides of the river are parallel.
    I labeled the original position as B and the second position as A
    AB = 75
    I labelled the reference point on the opposite side as P
    I also labelled PC as the perpendicular distance between the 2 sides
    angle ABP = 110°
    exterior angle A = 150°
    So, BAP = 30° and angle APB = 40°

    in triangle ABP, using the sine law:
    BP/sin30 = 75/sin40
    BP = 75sin30/sin40 = ....
    Now in the right-angled triangle BCP
    sin70° = CP/BP
    CP = BPsin70, but you know BP, so ......

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  3. Tan110 = d/X.
    d = x*Tan110.
    .
    Tan150 = d/(x-75).
    d = (x-75)*Tan150.

    x*Tan110 = (x-75)*Tan150.
    x = (x-75)*Tan150/Tan110,
    x = (x-75)*0.21,
    x = 0.21x - 15.75,
    X = -20 Ft.

    d = x*Tan110 = -20*Tan110 = 55 Ft.

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  4. 1.) B - 96.68
    2.) B - 58.34 ft
    3.) B - 39
    4.) D - 48.4
    5.) D - 70.0
    6.) B - 95
    7.) C - 169.3 ft
    8.) *not sure if this is correct*
    sin10 = AB/x
    0.1736 = 500/x
    x = 500/0.1736
    x = 2880.18

    amosc :) - annee-mrie

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  5. i wanna kiss you and give you money annee i love u girl

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  6. I Gotchu

    All your answers are correct.
    But the last one is not. The correct answer and work would be
    Cos 80 = 500/x
    x = 500/cos 80
    x = 2879.4 meters

    This work is from brainly, I wish it would allow me to put the link,
    But is doesn't. Good luck on everyone's quiz.

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  7. I wanna know where you got the 80 from

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  8. Or I might just have a different question

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  9. Yea I was right. It is a different question. For honors only I think. I put;
    cos(10) = 500/x
    x = 500/cos(10)
    x = 507.7

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  10. Annie really said "amosc" lol. Ty for the answers, appreciate it.

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