A string is wound around a cone with base radius 1 cm and slant height 10 cm. The string is wrapped in a helical manner, as if a strand of lights decorating a Christmas tree, except that it is pulled taut around the cone so as to have no local "wiggles", thus making what is known mathematically as a geodesic curve on the surface of the cone.

What is the maximum length of such a spiral around the cone before it starts to intersect itself?

(In reply to computer solution by Charlie)

With you all the way...

..and I notice that your expression for the length
10(cosA + sinA tan18) can be written as
10(cosA + sinA sin18/cos18)
= 10(cosA cos18 + sinA sin18)/cos18
= 10 cos(A - 18) /cos18
which clearly takes its maximum value of 10/cos18 (= 10.5146) when A = 18 as your computer suggested.

It's also interesting that, since the 'vertex' angle in the right angled triangle is 72 degrees, the string reaches its highest point after exactly 2 revolutions and meets itself after a further half revolution.

Much fun. 

Posted by Harry on 2009-08-06 21:07:15