2008 AIME I Problems/Problem 12

Problem

On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance form the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when is divided by .

Solution

Let be the number of car lengths that separates each car. Then their speed is . Let a unit be the distance between the cars (front to front). Then the length of each unit is .

Hence, we count the number of units that pass the eye in an hour: . We wish to maximize this.

Observe that as gets larger, gets less and less significant, so we take the limit as approaches infinity

Now, as the speeds are clearly finite, we can never actually reach full units. However, we only need to find the number of cars. We can arbitrarily increase their speed so that the camera stops (exactly one hour goes by) after the car part of the th unit has passed, but not all of the space behind it. Hence, cars is possible, and the answer is .