# 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.

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.