A couple of weeks ago I had to drive my son and his teammate to a swim meet about 60 miles away. We had to get up at about 5 in the morning; none of us having enough sleep. The boys dozed off almost immediately after settling in the car.
On long drives I made it a habit to talk math, so I prepared a few problems to discuss from the night before. But this time I had to rack my brain all by myself, if only to fight away drowsiness. The warm-up problem was simple [Math Circles, 1.18]:
The product of 22 integers is equal to 1. Show that their sum cannot be zero.
I had planned to modify it successively to
The sum of 25 integers is 0, can their product be 1?
The sum of 25 real numbers is 0, can their product be 1?
The sum of 25 rational numbers is 0, can their product be 1?
As the boy was sleeping, this had to wait for another trip. Meanwhile, I began to study the road.
As I said, we took an early drive and the road was pretty much void of traffic, not like the one below.
I could choose to drive in any lane or change lanes any time I wanted to, without arousing anybody's ire. The three-lane road was turning this way and that way all the time. How much could I gain by switching the lanes so as to always drive on the inner lane of a bend? Not that I did that, but thinking of the possibility was amusing and kept me awake.
My estimate for the width of a lane was 4 m. (Later I found that the minimum lane width on a freeway is specified as 12 ft, or about 3.66 meters.) This is also the distance between the center lines of two neighboring lanes. Assume the road makes 20° turns for every mile. On a two-lane road and a 60 mile drive, how much would I save always driving on the locally shortest lane?
For my purposes the turns add up even if they turn in opposite directions such that on a 60 mile drive the total turn would be 60×20° = 1200°, which is about 1200/360 ≈ 3.67 full revolutions.
The question is obviously related to the ages old quandary about one rope stretched over the equator and another over, say, a basketball. How much they should be lengthened if I want to move each 1 m away from the surface they are on. The answer, of course, is 2π in both cases. All that matters is how much the radius changes not the value of the radius itself. So, assuming I will be driving on a center line, the radius of the turn will change by 4 meters. 3.67 full revolutions with the radius of 4m amount to
Well, I finished the calculations at about the time we reached our destination - tells you how drowsy I was. We've been lucky there were no cars to bump into.