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Problem of the Week
Problem A and Solution
Old Faithful

Problem

Old Faithful is a geyser in Yellowstone National Park. It is so named because it was believed that it erupted every \(60\) to \(90\) minutes all day long. Assuming that Old Faithful erupts at \(12\) midnight and then erupts every \(60\) to \(90\) minutes after the last eruption, answer the following questions.

  1. After the first eruption at \(12\) midnight, what is the minimum number of eruptions you could see until up to and including \(12\) midnight the next night?

  2. After the first eruption at \(12\) midnight, what is the maximum number of eruptions you could see until up to and including \(12\) midnight the next night?

Solution

  1. One way to solve this problem is to make a timeline.

    We would see fewer eruptions if the time between eruptions is longer. The longest gap is \(90\) minutes. Therefore, the fewest number of eruptions will occur if the time between each eruption is \(90\) minutes.

    A 24 hour timeline starting and ending at midnight. A first arrow has its tail at the start of the timeline and its head at a point 90 minutes later. A second arrow, identical to the first, starts where the first arrow ended and ends 90 minutes later at 3 am. More arrows, with the next always starting where the previous one ended, cover the timeline from midnight to midnight.

    From this timeline we can count the ending point of each of the arrows in the diagram and see that, after the first eruption at \(12\) midnight, there would be \(16\) eruptions if they happened every \(90\) minutes. Note that the last eruption would be at exactly midnight on the next night.

    Alternatively, we might notice that \(90\) minutes is equal to \(1\frac{1}{2}\) hours, and \(180\) minutes (or two geyser eruption intervals) is equal to \(3\) hours. Then we can make a table keeping track of how many eruptions take place over time.

    Eruptions \(2\) \(4\) \(6\) \(8\) \(10\) \(12\) \(14\) \(16\)
    Hours Elapsed \(3\) \(6\) \(9\) \(12\) \(15\) \(18\) \(21\) \(24\)

  2. We would see more eruptions if the time between eruptions is shorter. The shortest amount of time between eruptions is \(60\) minutes. Since \(60\) minutes is equal to \(1\) hour, and there are \(24\) hours from \(12\) midnight until \(12\) midnight the next night, then the maximum number of eruptions we could see is \(24\) eruptions.