In our last post, we explored a curious problem about which lane to choose in traffic and found some surprising correlations to math. However, when we last saw our equation we were stuck with too many variables, so now we return to address that problem of how to find the measure of the two central angles.
In the real world, the answer to this question might be to get out some sophisticated equipment to measure this angle. That won’t be the approach that we’ll take because on a standardized test there isn’t a way to acquire new information. You have only what you’re given on the page. That can be an advantage, however, because limiting the number of potential inputs means that the same types of tricks and techniques you use on one problem are likely to be applicable to things you see in the future. In this case, the trick we’ll use is making the angles the same.
As I said before the measure of the central angle leading to the outside ring will be smaller because the cars are not backed up quite as far in that lane. However, in order to take advantage of the information we have, let’s use the same angle for both arcs, and compensate for the open space in the outer arc later. If we do that, we get:
*\frac{\theta}{360})
Since we now have our terms lining up in a way that will allow them to cancel out, we just need to deal with that shorter outer arc. The measure of the outer arc will be shorter than the equation listed above by the effective length of one car, that we estimated above at 20 feet. So, feeding that into our equation we get:
*\frac{\theta}{360}-20\right])
*\frac{\theta}{360}+20)
Since we set this up as the inside lane minus the outside lane, any positive difference would mean that the inside lane had to travel a longer distance, and thus the outside lane would be the better choice, or vice versa. Calculating the break-even point (where either lane would have to travel the same distance) is:
Solving gets us:
So, if the measure of the central angle that cuts the arc is greater than 96 degrees, the inner lane is the wise choice. However, if the measure of the central angle that cuts the arc is less than 96 degrees, the outer lane is best. Having traveled this road many times, I know that this long on-ramp cuts an arc longer than 96 degrees, and thus I (luckily, since I couldn’t do all these calculations on the fly) made the correct choice in taking the inner lane.
Next time, we’ll wrap this up by asking what all of this means.
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