perplexus.info :: Shapes : Your Yin, My Yang

Your Yin, My Yang (Posted on 2008-01-19)

Consider a cylinder having the two major halves of the Yin-yang symbol as the profile of two moulded pieces of differently coloured woods.

Allow one of these pieces to be obliquely cut from A to B as in the illustration.

What percentage by volume of the whole cylinder do each of these two parts represent?

The lower part of the illustration attempts to portray a receding void on the left and having cylindrical dimensions against a similar protuberance on the right, and of the same scale.

Also consider the lower section of the graphic to be an 'abbreviated' cross-section of the top, but rotated through 90°. And, if it wasn't obvious, the small black and white circles are to be dismissed from your thoughts.

See The Solution Submitted by brianjn

Rating: 4.0000 (1 votes)

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Solution

| Comment 1 of 2

The cylinder volume is V= Area of base * height = Ah, where A = pi*r^2 and V = (pi * r^2 *h)

The total volume of the sliced cylinder is V/2 = pi * r^2 *h/2
=1/2 (pi * r^2 * h)

Divide the solid into 4 parts: A cylinder near point A made of only white wood. A cylinder near point B made of only black wood. Two odd shaped solids (one of black, one of white), that occupy all "sides" of the larger cylinder.

The volume of the white wood cylinder is Vw = Aw * (h + h/2)/2
Aw = pi*(r/2)^2 = (pi/4) * r^2
Vw = pi/4 *r^2 * 3h/4 = 3/16 * (pi * r^2 * h) = 3V/16

The volume of the black wood cylinder is Vb = Ab * (h/2)/2
Ab = Aw = (pi/4) * r^2
Vb = pi/4 * r^2 * h/4 = 1/16 * (pi * r^2 * h) = V/16

The volume of the sides (combined) make up the remainder
Vs = V/2 - Vw - Vb
Vs = V/2 - 3V/16 - V/16 = V/4

The volume of the black side is Vbs = V/8

The volume of the white side is Vws = V/8

Thus the volume of the black is Vb + Vbs = V/16 + V/8 = 3V/16
= 18.75% of V

The volume of the white is Vw + Vws = 3V/16 + V/8 = 5V/16
= 31.25% of V

Edited on January 24, 2008, 3:02 pm

Posted by Leming on 2008-01-24 14:59:30

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