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Thread: plane geometry

  1. #1
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    plane geometry

    State the rule which connects BT, EA, and BA.

    this is a topic of plane geometry and we were given a directed investigation 'How far can you see the horizon?' this is it....the circle represents a cross-section of the Earth. imagine you are in a balloon at B. BT shows the furthest point, T, on the Earth's surface that you could see. imagine that you have just dropped a very heavy, sharp object from your balloon. the object, hits the ground at E, directly below you, boring its way through Earth coming to a stop just as it reaches the antipodal point A.

    here's a diagram: i have an attachment and its called doc. please someone answer this, its due monday and its worth 25% of my grade.
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  2. #2
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    Hello, TWoods!

    How far can you see to the horizon?

    The circle represents a cross-section of the Earth.
    Imagine you are in a balloon at $\displaystyle B.$
    $\displaystyle BT$ shows the furthest point, $\displaystyle T$, on the Earth's surface that you could see.
    Code:
                    B
                    o
                    |\
                    | \
                   h|  \
                    |   \
                    |    \
                  * * *   \
              *     |     *\ T
            *       |       o
           *       R|     *  *
                    |   *R
          *         | *       *
          *         o         *
          *         O         *
    
           *                 *
            *               *
              *           *
                  * * *

    The balloon is at $\displaystyle B,\;h$ miles above the Earth.
    The center of the Earth is $\displaystyle O$; the radius is: $\displaystyle OT = R$


    The angle at $\displaystyle T$ is 90.

    Using Pythagorus on $\displaystyle \Delta BTO\!:\;\;BT^2 + OT^2 \:=\:BO^2 \quad\Rightarrow\quad BT^2 \:=\:BO^2 - OT^2$

    . . $\displaystyle BT^2 \;=\;(R+h)^2 - R^2 \;=\;2Rh + h^2$


    Therefore: .$\displaystyle d = BT$, the distance from the balloon to the horizon

    . . is given by: .$\displaystyle \boxed{d \;=\;\sqrt{2Rh + h^2}}$ miles

    where: .$\displaystyle h$ = height of balloon, $\displaystyle R$ = radius of the Earth (both in miles).

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