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.

	Submitted by brianjn
Rating: 4.0000 (1 votes)
Solution:	(Hide)
Creating a "concrete" visual of this may seem horrendous. Once that is done there is really little mathematics left. Consider this reckless diagram as the top view of the Yin-yang cylinder being composed of 3 cylinders; the larger has two "sleeves" to carry two cylinders which are exactly half the diameter of the larger. A // \\ | \ / | | / \ | \\ // B By area, each part of the diagram is 1/4 of the larger circle, which does translate to 1/4 of the volume of the cylinder. A 1 2 |\ | | | \ | | | \| | | |\ | | | \ | | | \|B Now one half of the larger cylinder sliced obliquely from A to B is 1/8 of the whole. Consider the diagram above to be the two smaller "sleeved" cylinders sliced obliquely from A to B. The lower triangle, under column 2 represents 1/4 of the volume of that cylinder, or 1/16 of the whole. The 1/8 of the outer plus the 1/16 represent 3/16 of the total volume. The compliment of this is therefore 5/16. The cylinder is therefore divided into four such that two parts are 18.75% and the others are 31.25%. If I have gone too fast for some, consider the small circles to be of radius "1" and the larger "2". Πr2 yields 1*Π and 4*Π. Since the two smaller cylinders are 1/4 of the volume then the two remaining parts of the "main" cylinder are 1/4 each. It is important to make an acknowledgment. Leming has gone to a lot of work to give the mathematics of this situation. In actual fact I did go through exactly the same process before I realised that there was an 'intuitive' solution, but that does depend a little upon past knowledge. Read Leming

	Subject	Author	Date
	re: Solution	brianjn	2008-01-25 03:16:05
	Solution	Leming	2008-01-24 14:59:30