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Math Help - tracking a satellite

  1. #1
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    tracking a satellite

    Tracking a satellite

    The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 50 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0 and 84.2, respectively

    A. How far is the satellite from station A?

    B. How high is the station above the ground.

    I'm lost on this problem. Can you help me? Thanks.
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  2. #2
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    Quote Originally Posted by Godzilla View Post
    Tracking a satellite

    The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 50 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0 and 84.2, respectively

    A. How far is the satellite from station A?

    use the law of sines

    B. How high is the station above the ground.

    \textcolor{red}{h = a\sin{B} = b\sin{A}}
    ...
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  3. #3
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    I tried using the law of sines but no matter how many times I try it, I get a different answer than the book. I was hoping someone could step by step through it and then maybe I could figure out what I'm doing wrong.
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  4. #4
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    "The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 50 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0 and 84.2, respectively. A. How far is the satellite from station A? B. How high is the station above the ground.
    "

    Assume that it is not a spherical plane.

    Solution: (Drawing not shown)

    Let C be the satelite position, drop a line perpendicular to line AB, the intersetion will be point D. Let h be the distance of line CD.

    consider triangle ACD: tan 87 = h/BD

    h = BD tan 87 ---- (1)

    consider triangle BCD: tan 84.2 = h/(BD + 50)

    h = (BD + 50) tan 84.2 ---- (2)

    Equate equations (1) and (2),

    BD tan 87 = (BD + 50)tan 84.2

    BD(tan 87/tan 84.2) = BD + 50,

    BD(tan 87/tan 84.2 - 1) = 50,

    BD = 50/((tan 87/tan 84.2) - 1),

    BD = 53.3 miles,

    Answers:

    B) If the height of the station above the ground is BD, then

    BD = 53.3 miles

    A) The distance from from point A to the satelite is AC,

    then we have sin 87 = BD/AC,

    AC = BD/sin 87 = 53.3/sin 87 = 53.4 miles
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  5. #5
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    Quote Originally Posted by Godzilla View Post
    Tracking a satellite

    The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 50 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0 and 84.2, respectively

    A. How far is the satellite from station A?

    B. How high is the station above the ground.

    I'm lost on this problem. Can you help me? Thanks.
    It is necessary to assume:
    1) The tracking stations are at the same distance above mean sea level.
    2) The Radius of the earth is 3957 miles (that is somewhere between the polar radius about 3950 miles and the equatorial radius of about 3963 miles).
    3) The angle of elevation given is from a "level" line and NOT from sighting the other station and reading the vertical angle.


    Curvature:
     \dfrac{50}{3957} \times \dfrac{180}{\pi} = 0.724 degrees

    Adjust to angle of elevation:
    0.724/2 = 0.362 degrees at each station

    If the satellite is between the two stations the angles are, from A: 87 + 0.362 , or 87.362 degrees, and B: 84.2 + 0.362, or 84.562 degrees

    If the satellite is NOT between the two stations the angle from A is : 87 - 0.362 , or 86.638 degrees, and the angle from B is unchanged: 84.2 + 0.362, or 84.562 degrees.



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