# Triangle Problem- Very Urgent

• October 17th 2012, 07:34 PM
Makaveli
Triangle Problem- Very Urgent
Three wooden equilateral triangles of side length 18 inches
are placed on axles as shown in the diagram to the right. Each axle
is 30 inches from the other two axles. A 144-inch leather band is
wrapped around the wooden triangles, and a dot at the top corner is Attachment 25269
painted as shown. The three triangles are then rotated at the same
speed and the band rotates without slipping or stretching. Compute
the length of the path that the dot travels before it returns to its
initial position at the top corner
• October 18th 2012, 12:22 AM
MathoMan
Re: Triangle Problem- Very Urgent
The height of the wooden triangles is $h=\frac{18 \sqrt{3}}{2}=9\sqrt{3}.$. Rotation of the wooden triangles makes the point travel along the circumscribed circle. The radius of that circumcircle is $r=\frac{2}{3} h=\frac{2}{3}\cdot 9\sqrt{3}=6\sqrt{3}.$ Around every triangle the point travels along two thirds of that circle. And in between the circles it has to travel $3\cdot \left(\frac{144}{3}-2\cdot 18\right).$ You do the math, and see if this makes any sense.
• October 18th 2012, 06:26 AM
Makaveli
Re: Triangle Problem- Very Urgent
Thanks, Yeah I can see the circumcircle. Could i not just use your circle formula to find the circumference of the circle, which would then be the distance the dot traveled?
• October 19th 2012, 12:41 PM
Makaveli
Re: Triangle Problem- Very Urgent
Could someone please figure this out and show me how they got the answer please? My test is on Sunday, I have to go into school to take it. Thanks so much.
• October 20th 2012, 03:18 PM
iuhaseou
Re: Triangle Problem- Very Urgent
This problem is from an ongoing competition, DO NOT REPLY!! This is problem 24/2/1 from the USA Mathematical Talent Search, and as it is very, very clearly stated in the rules:
Quote:

Participants must not discuss the problems with others before the deadline for solution submission. Participants may use resources such as books or the internet to do mathematical research to try to solve the problems. Participants may not use 'live' online help, e.g. they may not ask for help with the problem on online forums or tutorial services. ... etc. ...