Question

Old Faithful Old Faithful is a geyser in Yellowstone National Park named for its regular eruption pattern. Past data indicates that the average time between eruptions is \(1 \mathrm{~h} 35 \mathrm{~m}\) (a) Suppose rangers log the first eruption on a given day at 12: 57 am. Using \(a_{1}=57,\) write a prediction formula for the sequence of eruption times that day in terms of the number of minutes after midnight. (b) At what time of day (e.g., \(7: 15 \mathrm{am})\) is the 10 th eruption expected to occur? (c) At what time of day is the last eruption expected to occur?

Short answer

(a) a_n = 57 + (n-1) * 95. (b) 3:12 pm. (c) 11:07 pm.

Step-by-step solution

Title - Convert time to minutes after midnight

Convert the average time between eruptions into minutes. Since the given time is 1 hour 35 minutes, the conversion is: 1 hour = 60 minutes So, 1 hour 35 minutes = 60 + 35 = 95 minutes.

Title - Identify first eruption time

The first eruption is logged at 12:57 am. Convert this time to minutes after midnight. Since there are 57 minutes past midnight, we have: a1 = 57 minutes.

Title - Write prediction formula

Using the information that each eruption is 95 minutes apart, the time for the nth eruption (an) can be given by the arithmetic sequence formula: a_n = a1 + (n-1)d where a1 = 57, d = 95, and n is the nth eruption. Therefore, the formula becomes: a_n = 57 + (n-1) * 95.

Title - Calculate the time of the 10th eruption

Substitute n = 10 into the formula: a_10 = 57 + (10-1) * 95 a_10 = 57 + 9 * 95 a_10 = 57 + 855 a_10 = 912 minutes after midnight. Convert this time back to standard time: 912 minutes = 15 hours 12 minutes. Therefore, the 10th eruption is at 3:12 pm.

Title - Calculate the time of the last eruption

Assuming the last eruption happens at midnight, which means the next day, thus, we need the total number of eruptions in one full day. The number of minutes in a day = 1440 Number of eruptions per day = Total minutes per day / Time between eruptions = 1440 / 95 ≈ 15.16 Since we cannot have a fraction of an eruption, we round down to 15 eruptions per day. Calculate the last eruption using n = 15: a_15 = 57 + (15-1) * 95 a_15 = 57 + 14 * 95 a_15 = 57 + 1330 a_15 = 1387 minutes after midnight. Convert this time back to standard time: 1387 minutes = 23 hours 7 minutes. Therefore, the last eruption of the day is at 11:07 pm.

Key concepts

arithmetic sequence

An arithmetic sequence is a list of numbers where each number is obtained by adding a fixed, constant value to the previous number. This fixed value is known as the 'common difference'.
In our case, the eruption intervals of Old Faithful geyser can be represented by an arithmetic sequence. The first eruption is at 12:57 am which translates to 57 minutes past midnight. Each subsequent eruption happens every 95 minutes.
Therefore, the arithmetic sequence formula we use is: a_n = a1 + (n-1)d Here, - a1 = 57 (first eruption time in minutes) - d = 95 (time interval between eruptions) This formula allows us to predict the minute of the day any eruption will occur.
For example, for the 10th eruption, we calculate: a_10 = 57 + (10-1) * 95 = 57 + 855 = 912 minutes.
This arithmetic sequence helps us predict the geyser’s behavior accurately.

time conversion

Time conversion is crucial when dealing with different units of time, such as converting hours and minutes into minutes after midnight. This is important in scenarios like tracking the intervals between geyser eruptions.
To convert a given time into minutes after midnight:

• First, convert the hours into minutes (1 hour = 60 minutes).
• Then, add the minutes to get the total minutes after midnight.

For example, 1 hour and 35 minutes converts to: 1 hour = 60 minutes total = 60 + 35 = 95 minutes For the specific geyser example: the eruption happens at 12:57 am, so it is 57 minutes after midnight.Breaking this down, each step becomes clearer:

• Hour to minute: 1 hour = 60 minutes
• Adding extra minutes: 60 + 35 = 95 minutes
• For 12:57 am, it is simply 57 minutes past midnight.

This straightforward method ensures we can manipulate time data precisely for various calculations.

geysers

Geysers are fascinating natural phenomena that periodically discharge water and steam into the air. The pattern of these eruptions can often be predicted, as is the case with Old Faithful in Yellowstone National Park.
Old Faithful is known for its regular eruption schedule, making it a popular subject for studying predictable natural events. The average time between its eruptions is 1 hour and 35 minutes, making it somewhat predictable for visitors.
Understanding the eruption pattern involves:

• Noting the first eruption time.
• Using arithmetic sequences to predict subsequent eruptions.
• Converting eruption times into a standard clock format for ease of understanding.

These geysers offer a great practical application of mathematical concepts, like arithmetic sequences and time conversion, allowing students to see real-world uses of these theoretical ideas.
Old Faithful's predictability stems from its underlying volcanic activity, which heats groundwater until pressure causes an eruption. This cycle repeats, with relatively consistent intervals, making arithmetic sequences a perfect tool for prediction.