Problem 81 - A Graphical Approach to Precalculus with Limits

Question

If a clock strikes the proper number of chimes each hour on the hour, how many times will it chime in a month of 30 days?

Step-by-Step Solution

Answer

The clock will chime 4680 times in a 30-day month.

Step 1: Calculate Chimes for One Day

A clock strikes the number of hours at the hour. For example, it strikes 1 time at 1 o'clock, 2 times at 2 o'clock, and so on up to 12 times at 12 o'clock. So, in one full day, the number of chimes can be calculated by summing these:\[1 + 2 + 3 + \ldots + 12\].

Step 2: Sum the Chimes for 12 Hours

Use the formula for the sum of an arithmetic series, which is \(S = \frac{n}{2} (a_1 + a_n)\), where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term, to calculate the sum:\[S = \frac{12}{2} (1 + 12) = 78\].

Step 3: Calculate Chimes for 24 Hours

Since the clock repeats the same sequence once during the day (AM and PM), you multiply the total number of chimes for one cycle by 2:\[78 \times 2 = 156\].

Step 4: Calculate Chimes for 30 Days

Multiply the total chimes for one day by the number of days in the month to get the total for 30 days:\[156 \times 30 = 4680\].

Key Concepts

Clock Chimes

Understanding how a clock chimes can help solve various mathematical problems. Each day, a typical clock strikes once for each hour it represents. That is:

• 1 time at 1 o'clock
• 2 times at 2 o'clock
• ... and so on until ...
• 12 times at 12 o'clock

This sequence completes in a 12-hour cycle, repeating twice in a 24-hour period (AM and PM). To find the total chimes in one day, we calculate the chimes from 1 to 12, which combines into an arithmetic series. Understanding this cyclical pattern is crucial for solving the exercise.

Sum of Series Formula

The sum of the chimes that occur in one cycle of the clock can be determined using the sum of series formula. The series, in this case, is an arithmetic series with uniformly increasing terms. Each number representing the hours. The formula to find the sum of any arithmetic series is:\[ S = \frac{n}{2} (a_1 + a_n) \]Where:

• \( n \) is the total number of terms in the series,
• \( a_1 \) is the first term,
• \( a_n \) is the last term.

In the context of this clock problem:

• \( n = 12 \)
• \( a_1 = 1 \)
• \( a_n = 12 \)

Solving this gives:\[ S = \frac{12}{2} (1 + 12) = 78 \]This means the clock chimes a total of 78 times over a 12-hour period.

Mathematical Problem Solving

Engaging in mathematical problem solving involves more than just calculating numbers. It requires understanding patterns and applying formulas creatively. With the clock chimes problem, we start by identifying the pattern of chimes over a 12-hour period and then using the sum of series formula to simplify the calculation.Once the total chimes for 12 hours was found, you need to remember that these occur twice in a full day (AM and PM). Thus, a key step is to multiply by 2:\[ 78 \times 2 = 156 \]Next, we extend the calculation over a 30-day period by multiplying the daily total by the number of days:\[ 156 \times 30 = 4680 \]This exercise exemplifies how breaking down a problem into smaller, manageable parts, and then utilizing arithmetic formulas, can effectively solve complex problems. By considering each component one step at a time, students can develop a deep understanding of the problem and derive the correct solution.