Belt Problem - Solution

Solution

Clearly triangles ACO and ADO are congruent right angled triangles, as are triangles BEO and BFO. In addition, triangles ACO and BEO are similar. Therefore angles CAO, DAO, EBO and FBO are all equal. Denoting this angle by, the length of the belt is

This uses the fact that the length of an arc = the radius × the measure of the angle facing the arc in radians.

To find we see from the similarity of triangles ACO and BEO that


For fixed P the length of the belt depends only on the sum of the radius values r1 + r2, and not on their individual values.

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