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Partitioning a unit disk?
Consider a disk of unit radius centered at the origin in the xy-plane. The disk is now divided into sixteen parts by the coordinate axes and the four lines x = +/- a, y = +/- a. Can "a" be chosen in such a way that all 16 pieces have identical areas? Justify your answer (and if it's "yes," find "a").
1 Answer
- AmyLv 76 years agoFavorite Answer
One of the 16 pieces is the square between (0,0), (0,a), (a,a), and (a,0). Its area is a^2. Its area must be 1/16 of the units circle's area, so a^2 = pi/16.
a = sqrt(pi/16).
The question is then whether each of the other sections also has area pi/16.
To the right of that square is a piece with corners (a,a), (a,0), (0,1), and (x,a) where that last point is on the circle, so x^2 + a^2 = 1, and therefore x = sqrt(1 - pi/16)
The area is greater than the area a * (x-a), which is a rectangle that fits inside the piece.
Throwing those numbers into a calculator, (x - a) > a.
This means the piece is bigger than the a^2 piece.
