I found the same solution: answer E (1/4, 1/pi). But I did it with a little more work! It's not easy to put my full explanation in this textbox... For my students I made the following explanation/answer:

same result as Henkie's but with a more geometric flavor to it:

http://bleaug.free.fr/8foxes/8foxes344.png

assume centroid of upper half circle (blue) is given: A(0,4/3pi) as hinted by anonymous, then we can construct centroids of red and green smaller half circles (Thales theorem).

Yin figure is equivalent to "blue-red+green". Yin centroid Z is the centroid of ABC triangle with respective weights 4, -1, 1. A(4)+B(-1) yields P(3), hence PA/PB=PZ/PC=AZ/BC=1/4 (Thales again). So: Zx=1/4 Zy=4/3pi-1/3pi=1/pi

A tip: the centroid of a semicircle is (4r/3pi) above its center. You know what I mean...

ReplyDeletex=[(-pi/8)(-0.5)+(pi/8)(0.5)]/(pi/2)=1/4

ReplyDeletey=[(pi/2)(4/(3pi))-(pi/8)(2/(3pi))+(pi/8)(-2/(3pi))]/(pi/2)=1/pi

jyu

I found the same solution: answer E (1/4, 1/pi). But I did it with a little more work! It's not easy to put my full explanation in this textbox...

ReplyDeleteFor my students I made the following explanation/answer:

http://home.kpn.nl/henkreuling/solutions8foxes/344_8foxes_solution.pdf

same result as Henkie's but with a more geometric flavor to it:

ReplyDeletehttp://bleaug.free.fr/8foxes/8foxes344.png

assume centroid of upper half circle (blue) is given: A(0,4/3pi) as hinted by anonymous, then we can construct centroids of red and green smaller half circles (Thales theorem).

Yin figure is equivalent to "blue-red+green". Yin centroid Z is the centroid of ABC triangle with respective weights 4, -1, 1. A(4)+B(-1) yields P(3), hence PA/PB=PZ/PC=AZ/BC=1/4 (Thales again). So:

Zx=1/4

Zy=4/3pi-1/3pi=1/pi

bleaug

Thank you all for these good solutions. Nice to see a geometric one as well. That's what we're looking for.

ReplyDeleteSo, here we say:

When a piece is removed from a whole, then the centroid is pushed out "linearly" and "proportionally".

When a piece is added to a whole, then the centroid is pulled in "linearly" and "proportionally".

Rest is simple math.

-8foxes