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Math Help - Chicago's Sears Tower

  1. #1
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    Chicago's Sears Tower

    The tallest building in the world is the Sears Tower in the city of Chicago. If the observation tower is 1450 feet above the ground, how far can a person see standing in the observation tower using the fact Earth's radius is 3960 miles?

    NOTE: 1 mile = 5280 feet.
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  2. #2
    Eater of Worlds
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    The Sear's tower is 1450/5280 miles tall.

    You have a triangle. Use Pythagoras.

    3960+(1450/5280)=3960+(145/528)=2091025/528

    \sqrt{(\frac{2091025}{528})^{2}-3960^{2}}=\frac{455\cdot\sqrt{2929}}{528}\approx{4  6.64} \;\ miles
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  3. #3
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    ok

    In other words, we have a right triangle where one leg = 1450 and the other leg = 5280. I would need to find c = hypotenuse, right?
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  4. #4
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    Hello, symmetry!

    You need a better diagram . . .


    The tallest building in the world is the Sears Tower in the city of Chicago.
    If the observation tower is 1450 feet above the ground,
    how far can a person see standing in the observation tower
    using the fact that Earth's radius is 3960 miles?
    Code:
                    *
                    | \
                   h|   \
                    |     \d
                  * * *     \
             *      |      *
           *        |       /*
          *        R|     /   *
                    |   / R
         *          | /        *
         *          *          *
         *                     *
    
          *                   *
           *                 *
             *             *
                  * * *

    The height of the tower is h.
    The radius of the earth is R.
    The distance from the top of the tower to the horizon is d.
    d and R form a right angle.

    From the right triangle: . d^2 + R^2\:=\:(h + R)^2
    . . which simplifies to: . d^2\:=\:2Rh + h^2\:=\:h(2R + h) [1]


    We are told that: . R = 3960 and h = \frac{1450}{5280} = \frac{145}{528}

    Substitute into [1]: . d^2\:=\:\frac{145}{5287}\left(2\cdot3960 + \frac{145}{528}\right) \:=\:2175.075417

    Therefore: . d \:\approx\:46.63 miles.

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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by symmetry View Post
    In other words, we have a right triangle where one leg = 1450 and the other leg = 5280. I would need to find c = hypotenuse, right?
    What you need here is a diagram, see below

    RonL
    Attached Thumbnails Attached Thumbnails Chicago's Sears Tower-gash.jpg  
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  6. #6
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    ok

    I want to thank everyone who took time out to answer this question, especially soroban and captainblack for their excellent diagrams of the Earth and triangle.

    A job well-done!

    I fully understand this question now.

    Thanks!
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