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 ;-).

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?!

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)

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agreed... requires brute force at some point.

ReplyDeleteFor 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

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

ReplyDeleteuse 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.

ReplyDeletedon'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

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.

ReplyDeleteLet 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

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

ReplyDeletehere 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.

ReplyDeletehttp://bleaug.free.fr/8foxes/fox353bis.html

(use arrows, page up/down or mouse wheel)

bleaug

Verified that 0.2928 is the answer.

ReplyDeleteI am getting two possible positions of E ,thus there may be two solutions: 0.2928 and 0.375

ReplyDeleteI have been tiny bit familiar of this your broadcast provided bright clear concept...

ReplyDeletegoldenslot

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Let's define as follows:

ReplyDelete* 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.