Problem Formulation

We aim to derive the relationship that governs the time taken to dial a phone number based on the arrangement of numbers on a phone keypad. The user is assumed to be right-handed and follows a sequential dialing pattern without pauses. Key considerations include:

• Key layout on the phone (standard 3x4 numeric keypad, where 1 is at the top left, 0 at the bottom middle)
• The distance moved by the finger between each digit
• The time per movement, which is influenced by the distance and speed

Variables and Assumptions

• \( N \): Number of digits in the phone number (e.g., 10 for a standard number)
• \( D_{i, i+1} \): Distance moved by the finger from digit \( i \) to \( i+1 \)
• \( v \): Average speed of finger movement between keys (assume consistent speed)
• \( t_{\text{press}} \): Time taken to press a single key
• \( T \): Total time taken to dial the phone number

Phone Keypad Layout and Distance Calculation

A standard phone keypad layout has keys arranged as follows:

         1    2    3
         4    5    6
         7    8    9
         *    0    #
    

Each key has a unique coordinate on this grid. For example, the digit 1 is at position \( (0,0) \), 2 at \( (0,1) \), and so on. We define the coordinates of each digit \( i \) as \( (x_i, y_i) \).

**Distance Between Digits**

The Euclidean distance between two consecutive digits \( i \) and \( i+1 \) is given by: \[ D_{i, i+1} = \sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2} \]

Total Dialing Time Calculation

The total dialing time \( T \) consists of two parts:

• The time to move between each consecutive pair of digits, based on the distance and speed \( v \).
• The time spent pressing each digit, assumed to be \( t_{\text{press}} \) for each key press.

Thus, the total dialing time \( T \) is:

\[ T = (N \cdot t_{\text{press}}) + \sum_{i=1}^{N-1} \frac{D_{i, i+1}}{v} \]

• \( N \): Total number of digits in the phone number
• \( D_{i, i+1} \): Distance between consecutive digits \( i \) and \( i+1 \)
• \( v \): Finger movement speed
• \( t_{\text{press}} \): Time to press each key

Example Calculation

Let's assume:

• A phone number with 10 digits
• Average finger movement speed \( v = 0.5 \) meters per second
• Press time \( t_{\text{press}} = 0.2 \) seconds
• Sample phone number: 1234567890, with coordinates:
  • 1: (0,0), 2: (0,1), 3: (0,2)
  • 4: (1,0), 5: (1,1), 6: (1,2)
  • 7: (2,0), 8: (2,1), 9: (2,2)
  • 0: (3,1)

Sample Calculation for Total Dialing Time

Using the phone number 1234567890 and the Euclidean distance formula, we calculate each \( D_{i, i+1} \) and sum them.

Steps:

1. \( D_{1,2} = \sqrt{(0 - 0)^2 + (1 - 0)^2} = 1 \) unit
2. \( D_{2,3} = \sqrt{(0 - 0)^2 + (2 - 1)^2} = 1 \) unit
3. \( D_{3,4} = \sqrt{(1 - 0)^2 + (0 - 2)^2} = \sqrt{5} \approx 2.24 \) units
4. Continue calculating for all pairs.

Assuming each unit distance is approximately 0.05 meters:

\[ T \approx (10 \cdot 0.2) + \sum_{i=1}^{9} \frac{D_{i, i+1} \times 0.05}{0.5} \]

Graphical Representation of Dialing Time vs. Number Length

The graph shows how dialing time increases as the number length grows, primarily due to the increased number of presses and distance traveled between digits.