The Peculiar Pillar (Posted on 2003-04-15)

	Submitted by Bryan
Rating: 4.0000 (6 votes)
Solution:	(Hide)
As stated in the problem, since the base of the pillar has 4 sides and the top has 3, two of the edges running the length of the pillar must meet at one of the corners of the triangle. Since the edge of the triangle opposite this corner is parallel to one of the edges of the bottom, every cross section of the pillar between the top and bottom is a trapezoid. To find the volume, we can integrate the trapezoidal cross sections along the height of the pillar. For a trapezoid with parallel sides B1 and B2 and height H, its area is A=.5(B1 + B2)H At the square base, B1 = B2 = H = 20cm At the top, B1 = 16, B2 = 0, & H = 8*sqrt(3) All three values change linearly from the base of the pillar to its top 200cm above. Let y = 0 at the base and y = 200 at the top. Then B1 = f(y) = 20 – y*4/200 = 20 – y/50 B2 = f(y) = 20 – y*20/200 = 20 – y/10 H = f(y) = 20 – y*20/200 + y*8*sqrt(3)/200 = 20 – y/10 + y*sqrt(3)/25 A = f(y) = .5(20 – y/50 + 20 – y/10)(20 – y/10 + y*sqrt(3)/25) A = y^2*(15 – 6*sqrt(3))/2500 + y*(4*sqrt(3) – 16)/5 + 400 To find volume V: V = integral[A dy] from 0 to 200 V = 1/3 *y^3*(15 – 6*sqrt(3))/2500 + .5*y^2*(4*sqrt(3) – 16)/5 + 400y ] from 0 to 200 V = 200^3*(15 – 6*sqrt(3))/7500 + .5*200^2*(4*sqrt(3) – 16)/5 + 80000 V = 48627.7 cm^3 Therefore the weight W of the pillar is W = 48627.7*2.8 = 136157.6 g = 136.16 kg Note: Whether the triangular top of the pillar is centered above the square base or not will have no effect on the pillar’s volume. v

	Subject	Author	Date
	Puzzle Answer	K Sengupta	2024-05-11 03:18:52
	Another Solution	P C	2004-02-04 20:36:39
	Hmmm...	Silly Jilly	2003-05-12 02:25:30
	I do not recall...	Tim Axoy	2003-05-11 13:15:21
	I do not recall...	Tim Axoy	2003-05-11 13:14:24
	Oh My God !!!!	Ravi Raja	2003-04-19 07:56:01
	solution	Tim Axoy	2003-04-18 12:54:46
	A Twist	Charlie	2003-04-15 05:02:49
	No Subject	Charlie	2003-04-15 05:02:05
	a solution	Hank	2003-04-15 04:22:19
	re: solution	Robin Gatter	2003-04-15 04:01:05
	My solution	Robin Gatter	2003-04-15 03:52:34
	solution	Charlie	2003-04-15 03:44:41