If a circle with a diameter of 4 inches rotates 1000°, what distance will any one spot on its edge actually travel?

1 Answer
May 6, 2018

#(100pi)/9#

Explanation:

First of all, note that #1000° = 360+360+280#. So, rotating #1000°# means to make two complete turns, plus an additional #280°# rotations. Let's deal with the complete rotations first: every spot on the edge will traver exactly one circumference with every turn.

Since the diameter is #4#, the radius is #2#, and thus we have

#C = 2pi r = 4pi#.

Then, a rotation of #280°# is trivially #280/360# of a whole turn. We can simplify the fraction to #7/9#.

Recap: #1000# degrees means two complete turns, plus an additional #7/9# of a turn:

#2+7/9 = 18/9+7/9 = 25/9#

With every turn, any spot travels exactly one circumference of distance, so the answer is

#25/9 C = 25/9 * 4pi = (100pi)/9#