Monday, November 5, 2012

Fox 353


Tony García‏ from Dominican Republic submitted this one.  It looks like it requires brute force.  Better than being idle I guess!  So enjoy...



The answer is not confirmed yet.

17 comments:

  1. agreed... requires brute force at some point.

    For those interested, exact value for sin(AEF) is ((261 - 19*sqrt(161))*sqrt((
    179745369 + 14093001*sqrt(161))/737237))/1500

    which approximately equals 0.292829, hence answer B.

    A "less brutal" question could have been: what is the value of Determinant(EF,EA)?

    bleaug

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  2. Please, show where the question require "brute force". Sorry, my mother language is Spanish: what do you mean by "brute force"?

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  3. use of less and less geometric arguments or abstract algebraic properties, more and more of cartesian coordinates or trigonometric formulae, and even, as a last resort, approx numerical equation solving.

    don't worry, we all like brute force once in a while.

    in this case, it starts once you've established that FB=... (I leave the exercise to the reader ;-).

    bleaug

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  4. Yes, this is true. "Brute force" means complete enumeration in combinatorial, or computing science. It may also be called as "exhaustive search" in different context. Here what we mean by it is: the solution require rigorous computations - as bleaug pointed out - solving high degree polynomials, quadratics, trigonometric, and numerical approximations... And ending up with long expressions.

    Let me make it more difficult to understand:
    See, there is a beauty if a solution says: x+sin(x)=1. It is a closed form formula but simple and aesthetic. We'd love it! We'll keep that way, don't even ask what x is.

    But when we say x=sqrt(12625-sqrt(98756-sqrt(3)))/1797. Then a certain level of nicety might have gone.

    Or am I wrong? That tedious expression might have its own merits and beauty? And may be not?!

    End of message... and confusion...

    8foxes

    ReplyDelete
  5. more and more of cartesian coordinates???? I have other approaches and I haven´t seen Cartesian coordinates..hehe

    ReplyDelete
  6. here is a solution using svg + mathml. It was designed with firefox, which supports both nicely. Chrome and Opera do well with svg but have problems with mathml. Not tested with ie.

    http://bleaug.free.fr/8foxes/fox353bis.html
    (use arrows, page up/down or mouse wheel)

    bleaug

    ReplyDelete
  7. Verified that 0.2928 is the answer.

    ReplyDelete
  8. I am getting two possible positions of E ,thus there may be two solutions: 0.2928 and 0.375

    ReplyDelete
  9. I have been tiny bit familiar of this your broadcast provided bright clear concept...



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    ReplyDelete
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  12. Let's define as follows:

    * b = height of the rectangle = CD
    * a = width of the rectangle = BC
    * c = BD = sqrt(CD^2 + BC^2)

    By using the inscribed circle:
    sqrt(1600/b^2 + b^2) = ((40/b) - (b-10))
    This leads to: b^2 - b - 40 = 0

    By taking the positive solution: b = 6.844

    Then:
    * a = 5.845
    * c = 9.000

    AE = sqrt(b^2 + (5.845)^2) = 7.442
    EF = sqrt((a*(1-5/c)-a/2)^2 +(b*(1-5/c))^2)) = 3.06
    AF = sqrt((a*(1-5/c))^2 + (b*(5/c))^2) = 4.605

    (1) Get the area of AEF by Heron's formula:

    s = (7.442 + 3.06 + 4.606) / 2 = 7.554
    S = sqrt(7.554*(7.554-7.442)*(7.554-3.06)*(7.554-4.606))
     = 3.348

    sin(AEF) = (3.348 * 2) / (7.442 * 3.06)
     = 0.294

    (2) Use the cosine formula to first get cos:

    cos(∠AEF) = (7.442^2 + 3.06^2 - 4.606^2)/
     (2 * 7.442 * 3.06)
     = 0.9558

    sin(∠AEF) = sqrt(1 - 0.9558^2)
     = 0.294

    (1) and (2) both well match.

    ReplyDelete
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