Before we get stuck and give up, let’s make sure we remember the task. Just because we can’t measure the lengths of the arcs, doesn’t mean that we can’t find the *difference* in lengths between the two arcs. Let’s take a look at the *real* equation we need to solve.

Which becomes:

First, let’s talk about our two radii. If we think about a radius extending from the center of the circle—wherever that may be—we know that it will reach the inner ring first. To get from the edge of the inner ring to the edge of the outer ring, that distance is simply the width of the lane, which we said before is 12 feet! So, we can substitute that back into our equation!

Before we do that however, let’s think about our other variable, theta. That variable measures the central angle that we need to measure. We know that both angles will end at the same place (where the traffic is allowed to merge on the freeway), but that the outer angle will be a bit smaller because of the one-car length difference in traffic. However, solving for the angular difference precisely seems extremely difficult. Perhaps we could estimate, but that’s about it. This poses yet another problem that we need to solve.

Here is where our equation sits now:

Can you figure out what the next step should be?