Question

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

Answer 1

`(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`