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C. 886. The wall of a building is decorated by a square with its circumscribed circle drawn. There is a semicircle drawn outwards on each side of the square as a diameter, so that the circular arcs form four crescent shaped figures. What is the length of the side of the square if the area of such a crescent is 1 m2?

(5 points)

Deadline expired.


Google Translation (Sorry, the solution is published in Hungarian only.)

Megoldás. Jelölje a négyzet oldalát x. Ekkor egy Hold alakú mező területe:

t=\frac{1}{2}\cdot\left(\frac{x}{2}\right)^2\cdot\pi-\left(\frac{1}{4}\cdot
\left(\frac{\sqrt2}{2}x\right)^2\pi-\frac{1}{2}\cdot\left(\frac{\sqrt2}{2}x\right)^2\right)=

=x2(\pi/8-\pi/8+1/4)=x2/4.

Vagyis 1=x2/4, ahonnan x=2 m.


Statistics on problem C. 886.
410 students sent a solution.
5 points:357 students.
4 points:8 students.
3 points:35 students.
2 points:7 students.
0 point:2 students.
Unfair, not evaluated:1 solution.


  • Problems in Mathematics of KöMaL, February 2007

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