Guys. It’s apparently going to break 100° today in New York. Seriously.

The prize this week for the first correct answer: You will awake in a bathtub of ice, and have no idea how you got there. Your first concern will be a suspicious scar on your abdomen, but that will quickly be replaced by relief that you are in a bathtub of ice, and not in New York City, where it is frikkin’ 100°.

A wheel is rolling in a straight line, without slipping, on a flat surface. Two points on the wheel have paint on them, and they are leaving spots on the surface as the wheel rolls (wheel is rolling from left to right in the figure). What is x?

UPDATE: You guys rock. Special congratulations to the anonymously sweltering newcomer, for getting it first. Great job.

Solution below the cut.

The key to getting this right is recognizing that when a wheel makes a revolution, if it’s not slipping, it travels exactly one circumference. So when this particular wheel makes spots on the surface 8 and 10 units apart, we know that the full circumference is 18 units. We also know that the next dot will be 8 away from the previous one, and that the angle we’re looking for corresponds to an arc length of 8.

From there, you simply must recognize the opportunity to use a part/whole ratio (see this post for more on how many difficult circle problems are actually ratio problems) to solve for x.

$\frac{central\:&space;angle}{360^{\circ}}=\frac{arc\:length}{circumference}$
$\frac{x^{\circ}}{360^{\circ}}=\frac{8}{18}$
$x=160$