Problem 116 - Precalculus

Question

Refer to the following: A common school locker combination lock is shown. The lock has a dial with 40 calibration marks numbered 0 to \(39 .\) A combination consists of three of these numbers (e.g., \(5-35-20\) ). To open the lock, the following steps are taken: \(\cdot\)Turn the dial clockwise two full turns. \(\cdot\)Continue turning clockwise until the first number of the combination. \(\cdot\)Turn the dial counterclockwise one full turn. \(\cdot\)Continue turning counterclockwise until the 2 nd number is reached. \(\cdot\)Turn the dial clockwise again until the 3 rd number is reached. \(\cdot\)Pull the shank and the lock will open. Given that the initial position of the dial is at zero (shown in the illustration), how many degrees is the dial rotated in total (sum of clockwise and counterclockwise rotations) in opening the lock if the combination is \(20-15-5 ?\)

Step-by-Step Solution

Answer

The dial is rotated a total of 1665 degrees to open the lock.

Step 1: Rotation for Initial Clockwise Turns

First, we perform two full clockwise rotations. Since each rotation covers 360 degrees, two full turns equal \(2 \times 360 = 720\) degrees.

Step 2: Rotation for First Number

After two full clockwise turns, continue to rotate clockwise to the first number, 20. Starting from 0, this requires a further 20 calibration marks. Each mark represents 9 degrees (since \(360 \div 40 = 9\) degrees per mark), so rotating to 20 adds \(20 \times 9 = 180\) degrees.

Step 3: Full Counterclockwise Turn

Now we rotate counterclockwise for one full turn (360 degrees) before setting it to the second number, 15. This involves one complete rotation, adding 360 degrees to the rotation total.

Step 4: Rotation for Second Number

Continuing counterclockwise to reach the second number, 15. Starting from 20, we move back to 0 (20 marks), and then to 15 (an additional 15 marks). In total, this is \(35 \times 9 = 315\) degrees.

Step 5: Rotation for Third Number

Finally, rotate clockwise to the third number, 5, from 15. Moving 10 marks clockwise (15 back to 5) equals \(10 \times 9 = 90\) degrees.

Step 6: Summing Total Rotation

Add all these rotations together for the total rotation: \(720 + 180 + 360 + 315 + 90 = 1665\) degrees.

Key Concepts

Clockwise Rotation

Clockwise rotation refers to turning an object in the same direction as a clock's hands move. In the context of a combination lock, when you turn the dial clockwise, you are following this path. Understanding how to count the rotation in degrees is essential for solving problems like opening a lock with a specific sequence.

When a full clockwise turn is completed, the dial covers 360 degrees. In the case of our lock, the first step requires two complete clockwise turns, meaning that the dial rotates 720 degrees (because 2 multiplied by 360 equals 720). This setup is important as it ensures the lock is reset before entering the combination.

If you continue to rotate clockwise directly to your first number after the initial turns, you'll need to account for those degrees as well. In our example, reaching the first number in your combination is another 180 degrees, as you would move over 20 marks at 9 degrees per mark, totaling up to 180 degrees. This brings the subtle yet crucial understanding of how these angles are added in a sequence.

Counterclockwise Rotation

Counterclockwise rotation is simply the opposite of clockwise movement. For combination locks, turning the dial counterclockwise often follows after initial clockwise rotations. This ensures that you can reset your dial and accurately reach the second number in your combination.

One important aspect is understanding how much your dial rotates in degrees when going counterclockwise. In this exercise, after reaching the first number, the dial should be turned one full counterclockwise circle, equivalent to 360 degrees. This rotation helps ensure the lock mechanism recognizes the intended stop points to fully secure its sequence.

After this full turn, you'll continue counterclockwise to your second combination number, further adding to the degrees of movement. From 20, moving back to 15 after circling past 0 adds 315 degrees (35 marks). Recognizing this sequence is crucial since needing to move marks backwards changes the total rotation.

Combination Locks

A combination lock, often found on school lockers, uses a sequence of numbers and rotations to secure its mechanism. Unlike conventional key locks, these rely on precise angular displacement of the dial to unlock.

The lock dial typically has numbers (ours has 40, from 0 to 39) marking its face. Opening involves understanding the rotational steps: beginning with multiple clockwise turns to reset, followed by specific moves to each combination number. These operations require awareness of degrees, especially when moving counterclockwise, as in reaching 15 from 20 after resetting through a full turn.

The challenge lies in carefully aligning both the direction and amount of each rotation. By realizing the angles and directions involved (720 degrees clockwise initially, followed by 180 and more clockwise turns for numbers, and counterclockwise adjustments like the 360 and 315 degrees in our problem), one can work toward mastery in understanding how angular displacement functions, making the art of using combination locks both functional and educational.