Answer:

1. As the length of time increases, the distance also increases. This is a linear relationship, where the distance traveled is directly proportional to the amount of time.

2. Using this pattern, if the runner travels 59 km in 2 hours, then we can determine how many kilometers he would have traveled in 8 hours by multiplying the distance traveled in 2 hours by 4 (since 8 is 4 times as long as 2). So, 59 km x 4 = 236 km.

3. You can find the distance without the aid of the table by recognizing the linear relationship between time and distance and using the concept of the constant rate of change. The mathematical statement to represent this relation can be written as: distance (km) = rate of change (km/hr) * time (hr). In this case, the rate of change is the speed at which the runner is traveling.

4. The mathematical operation applied in this case is multiplication. The constant number involved is the rate of change or the runner's speed. The process discovered is that the distance traveled is determined by multiplying the rate of change (speed) by the amount of time. This reflects the concept of distance being equal to speed multiplied by time, a fundamental principle in the study of linear relationships in physics and mathematics.

Answer:

1. **What happens to the distance as the length of time increases?**

- The distance increases as the length of time increases. This is evident in the pattern where, as time (hours) increases, the corresponding distance (kilometers) also increases.

2. **Using this pattern, how many kilometers would he have traveled in 8 hours?**

- To find the distance for 8 hours, you would look at the pattern in the table. Unfortunately, the provided table is incomplete. If you have the correct values, you can simply find the distance corresponding to 8 hours.

3. **How will you be able to find the distance (without the aid of the table)? Write a mathematical statement to represent the relation.**

- If there's a consistent pattern, you can use a linear equation to represent the relation. Assuming a linear relationship, the mathematical statement would be in

[tex] distance \: = m \: x \: times + b \: where \: ( m ) is \: the \: slope \: and \: ( b) \: is \: the \: y-intercept.[/tex]

4. **What mathematical operation did you apply in this case? Is there a constant number involved? Explain the process that you have discovered.**

- The mathematical operation involved is likely multiplication, as the distance increases with time. If there's a constant rate of change, it would be represented by the slope (m) in the linear equation. The process involves determining the consistent rate at which distance changes with time. If there's a constant number involved, it would be the y-intercept (b).

Without the complete table values, it's challenging to provide specific details, but these are general approaches to understanding the relationship between time and distance.

Step-by-step explanation:

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