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Math Help - Orbit coverage...

  1. #1
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    Orbit coverage...

    I'm not really sure what this question is asking of me. It says:

    A satellite has a geosynchronous orbit(it stays over the same point on the Earth) and broadcasts cable television from 100 mi. above the surface of the Earth. Assume taht the Earth is a sphere with radius 4000 mi, and find the arc length of coverage area for the cable television satellite on the Earth's surface.

    THANKS!!!!
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  2. #2
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    draw a circle to represent the earth.

    place a point (the satellite) above the earth's surface. (not very far, consider your scale of miles)

    draw two lines tangent to the earth that pass through the point.

    from the earth's center, draw lines to the two points of tangency on the earth's surface and one to the satellite point.

    two congruent right triangles are formed. using one of them ...

    hypotenuse = 4100 miles

    long leg = earth's radius = 4000 miles

    angle at earth's center = \arccos\left(\frac{4000}{4100}\right) = .2213 \, rad

    central angle that intercepts the satellite's transmission arc is double that angle ... \theta = 0.4426 \, rad

    now, remember this equation? ... s = r\theta
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