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Math Help - The ladder

  1. #1
    Senior Member TriKri's Avatar
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    The ladder

    A 10 meters long ladder is leaned against a wall and a 1 meter times 1 meter big box, as the picture illustrates. The width of the ladder is negligible. How high up on the wall does the ladder reach?

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  2. #2
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    Look at the similar triangles.
    (1 Blue dot and 2 red dots) and the triangle (1 Blue dot and 2 black dots).

    Thus,
    \frac{h-1}{1}=\frac{h}{w}

    And, by Pythagorean theorem,
    w^2+h^2=100

    Now you can solve
    Attached Thumbnails Attached Thumbnails The ladder-picture5.gif  
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  3. #3
    Senior Member TriKri's Avatar
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    This gives me a fourth order equation. I don't know how to solve them.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by TriKri View Post
    This gives me a fourth order equation. I don't know how to solve them.
    Take hackers equations and turn them into an interative scheme:

    <br />
w_n=\frac{h_n-1}{h_n}<br />
    <br />
h_{n+1}=\sqrt{100-w_n^2}<br />

    Now put h_0=9.9, h_1=9.93794, h_2=9.93799 , h_3=9.93799 .

    So h \approx 9.93799

    (we choose an initial guess of h=9.9 by doing a rough drawing of the
    arrangement and measuring the approximate height, but a rough guess of 9 will have
    added just one more iteration to get this level of accuracy)

    The scheme also produces an approimate value for w\approx 1.11188

    RonL
    Last edited by CaptainBlack; December 8th 2006 at 02:21 PM.
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  5. #5
    Forum Admin topsquark's Avatar
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    I would like to note that there is technically another solution: h = 1.11188 m, which the diagram (sort of) rules out. In this case the ladder is nearly horizontal rather than vertical. (I rarely take diagrams as being "to scale" or in the correct orientation unless I have to.)

    -Dan
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  6. #6
    Senior Member TriKri's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Take hackers equations and turn them into an interative scheme:

    <br />
w_n=\frac{h_n}{h_n-1}<br />
    <br />
h_{n+1}=\sqrt{100-w_n^2}<br />

    Now put h_0=9.9, h_1=9.959509, h_2=9.959454, h_3=9.959454.

    So h \approx 9.959454

    (we choose an initial guess of h=9.9 by doing a rough drawing of the
    arrangement and measuring the approximate height, but a rough guess of 9 will have
    added just one more iteration to get this level of accuracy)

    RonL
    Nice! But that's cheating!
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by TriKri View Post
    Nice! But that's cheating!
    In what way, I could have guessed the solution anyway I wanted.

    By the way for some reason the iterative solution appears to be wrong,
    that I will have to look into

    Fixed now. Note the value of w produced is topsquarks other solution.

    RonL
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  8. #8
    Super Member malaygoel's Avatar
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    I solved it without using numerical methods.
    It is a bit tricky.

    The ladder is 10m long and let at the point of contact with the box, it is divided into x(the upper part) and y(the lower part).{x+y=10}
    You see that
    \frac{1}{x^2}+\frac{1}{y^2}=1
    x^2 +y^2=x^2y^2
    (x+y)^2=(xy)^2+2xy
    Let xy=b
    100=b^2+2b
    b^2+2b-100=0
    since b is positive
    b(=xy)=\sqrt{101}-1
    x(10-x)=\sqrt{101}-1
    solving for x, we get
    x=5\pm \sqrt{24-\sqrt{101}}
    Since we are looking for maximum height,
    we have
    x=5+ \sqrt{24-\sqrt{101}}
    h=\sqrt{x^2-1}+1

    Keep Smiling
    Malay
    Last edited by malaygoel; December 27th 2006 at 08:36 PM.
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  9. #9
    Senior Member TriKri's Avatar
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    Wow, impressive! How did you know you should solve it like this? Is it some kind of standard equation in some branch of mathematics? I'm just wondering because I haven't read enough with math to be able to solve it myself... ^_^
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  10. #10
    Super Member malaygoel's Avatar
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    Quote Originally Posted by TriKri View Post
    Wow, impressive! How did you know you should solve it like this? Is it some kind of standard equation in some branch of mathematics? I'm just wondering because I haven't read enough with math to be able to solve it myself... ^_^

    It just striked me.
    I haven't studied fourth order equations, was trying to use the cube(its all sides are equal)

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  11. #11
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    That's some ladder problem. I didn't know it could get this hard :-P Man, my math class is wimpy XD
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  12. #12
    Senior Member TriKri's Avatar
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    Quote Originally Posted by benzi455 View Post
    That's some ladder problem. I didn't know it could get this hard :-P Man, my math class is wimpy XD


    Have you showed them malaygoel's solution?
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  13. #13
    Senior Member TriKri's Avatar
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    Quote Originally Posted by malaygoel View Post
    It just striked me.
    I haven't studied fourth order equations, was trying to use the cube(its all sides are equal)

    Keep Smiling
    Malay
    The cube? I haven't heard of that before, but not strange, regardning what I said before...

    I was trying to make the equation more general by puting the sides of the bow to a and b, but I couldn't solve it you'r way since I couldn't get only xy left in the equation and no x or y. But I'm still impressed. Is this the way to solve third order equations?
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  14. #14
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    Quote Originally Posted by TriKri View Post
    Is this the way to solve third order equations?
    Yes!

    It was first made by Italian mathematicians: Tartalia, Cardano. I think they kept it secret.

    I have memorized the original method for solving 3rd order equations . And can up with my own which is simpler but not always works, 2 nights ago when I was falling asleep I was thinking about a techinique that will make my solution always workable did not try it yet.

    Maybe, I will post it for thee.
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  15. #15
    Super Member malaygoel's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Yes!

    It was first made by Italian mathematicians: Tartalia, Cardano. I think they kept it secret.

    I have memorized the original method for solving 3rd order equations . And can up with my own which is simpler but not always works, 2 nights ago when I was falling asleep I was thinking about a techinique that will make my solution always workable did not try it yet.

    Maybe, I will post it for thee.
    Trikri!!
    i mistyped it.
    I want to say I was trying use the square.
    i don't know how to solve cubic equations.

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    Malay
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