Problem 115 - 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 \(35-5-20 ?\)

Step-by-Step Solution

Answer

The dial is rotated 1800 degrees in total to open the lock.

Step 1: Calculate degrees for the two full clockwise turns

Each full turn of the dial represents 360 degrees. Since the first instruction is to turn the dial clockwise two full turns, you rotate the dial by \(2 \times 360 = 720\) degrees clockwise.

Step 2: Calculate degrees for first number clockwise turn

After completing two full turns, continue turning clockwise until the first number of the combination, which is 35. From position 0 to 35 is an additional \((35 - 0) \times 9 = 315\) degrees clockwise because each numbered mark on the dial equals 9 degrees \((360 / 40)\).

Step 3: Calculate degrees for one full counterclockwise turn

Next, you turn the dial one full turn counterclockwise. This consists of a 360-degree turn.

Step 4: Calculate degrees for second number counterclockwise turn

Start from 35 and continue counterclockwise to reach the second number, 5. From 35 to 5 counterclockwise is \((35 - 5) \times 9 = 270\) degrees counterclockwise. Add the full turn and the additional rotation: \(360 + 270 = 630\) degrees counterclockwise.

Step 5: Calculate degrees for third number clockwise turn

Starting from 5, turn the dial clockwise to reach the third number, 20. From 5 to 20 is \((20 - 5) \times 9 = 135\) degrees clockwise.

Step 6: Total degree rotation calculation

Add up all the clockwise and counterclockwise rotations:- Total clockwise: \(720 + 315 + 135 = 1170\) degrees.- Total counterclockwise: 630 degrees.- The net total is \(1170 + 630 = 1800\) degrees.

Key Concepts

Angle Measurement

Angle measurement is crucial when dealing with combination locks. Each number on the dial corresponds to an angle increment. In the case of many standard locks, the dial has 40 marks. This means each increment between those numbers represents a division of the full 360-degree circle of the dial.

To determine the angle for each increment:

• Divide the full circle, 360 degrees, by the number of marks. For a dial with 40 marks, this amounts to 9 degrees per mark \( \left( \frac{360}{40} = 9 \right) \).
• Once you understand that each notch represents 9 degrees, you can calculate the angle by simply multiplying the mark number by 9.

Understanding these measurements helps us translate numbers on a dial to precise movements. Whether you're calculating several full rotations or smaller movements to a specific number, knowing the angles ensures accuracy.

Clockwise and Counterclockwise

The concepts of clockwise and counterclockwise revolve around the direction of rotation. Clockwise refers to the rotational direction towards the right, akin to the forward motion of clock hands. Conversely, counterclockwise is the opposite direction, toward the left, against the clock’s usual motion.

In lock mechanics, the difference matters:

• Clockwise movement involves turning the dial to the right, easing forward.
• Counterclockwise rotation involves turning the dial to the left, rolling back.

Understanding which direction to turn is vital in opening combination locks correctly. Each directional movement is tied to specific instructions. Misinterpreting clockwise and counterclockwise can result in errors, needing a restart of the process.

Lock Mechanics

Lock mechanics are the practical application of understanding our dial's rotation, direction, and sequence to successfully open the lock.

Here are some essential points:

• Starting with the initial two full clockwise rotations sets the lock's internal gears to a standard position, preventing accidental opening from half-turns.
• The sequence of clockwise and counterclockwise movements ensures the lock recognizes the correct combination.
• Each turn engages different internal gates within the lock, progressing its readiness to open.

Mastering these mechanics doesn't just rely upon memorizing a sequence, but also understanding how each movement aligns internal components to release the shank.