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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

Re: Triangle Problem- Very Urgent

The height of the wooden triangles is $\displaystyle h=\frac{18 \sqrt{3}}{2}=9\sqrt{3}.$. Rotation of the wooden triangles makes the point travel along the circumscribed circle. The radius of that circumcircle is $\displaystyle r=\frac{2}{3} h=\frac{2}{3}\cdot 9\sqrt{3}=6\sqrt{3}.$ Around every triangle the point travels along two thirds of that circle. And in between the circles it has to travel $\displaystyle 3\cdot \left(\frac{144}{3}-2\cdot 18\right).$ You do the math, and see if this makes any sense.

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?

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.

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. ...