Breaking Down the Equation

Oct 13, 2025 by recoveryshake.com 116 views

Max rides his scooter toward Kim and then passes her at a constant speed. The distance between Max and Kim $t$ seconds after he started riding his scooter is given by $d=|150-9t|$ .

In this problem, we are given an equation that represents the distance between Max and Kim at any given time $t$ . The equation is $d=|150-9t|$ . To understand what the 150 in the equation represents, we need to analyze the equation and its components.

Breaking Down the Equation

The equation $d=|150-9t|$ consists of two main components: the absolute value expression $|150-9t|$ and the variable $t$ , which represents time in seconds. The absolute value expression is a mathematical function that returns the distance between Max and Kim at any given time $t$ .

What is the Absolute Value Expression?

The absolute value expression $|150-9t|$ is a mathematical function that returns the distance between Max and Kim at any given time $t$ . The absolute value function is defined as:

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

In this case, the absolute value expression $|150-9t|$ returns the distance between Max and Kim at any given time $t$ . The expression $150-9t$ represents the difference between the initial distance between Max and Kim and the distance traveled by Max at time $t$ .

What does the 150 in the equation represent?

The 150 in the equation represents the initial distance between Max and Kim. This is the distance between them when Max starts riding his scooter. The equation $d=|150-9t|$ indicates that the distance between Max and Kim changes over time, with Max moving towards Kim at a constant speed.

Understanding the Initial Distance

The initial distance of 150 represents the starting point of the problem. It is the distance between Max and Kim when Max starts riding his scooter. This distance is a fixed value that does not change over time.

Conclusion

In conclusion, the 150 in the equation $d=|150-9t|$ represents the initial distance between Max and Kim. This is the distance between them when Max starts riding his scooter. The equation $d=|150-9t|$ indicates that the distance between Max and Kim changes over time, with Max moving towards Kim at a constant speed.

Further Analysis

To further analyze the equation, we can consider the following:

Distance traveled by Max: The expression $9t$ represents the distance traveled by Max at time $t$ . This distance is subtracted from the initial distance of 150 to obtain the current distance between Max and Kim.
Constant speed: The equation $d=|150-9t|$ indicates that Max moves towards Kim at a constant speed. This means that the distance between Max and Kim changes linearly over time.
Absolute value: The absolute value expression $|150-9t|$ ensures that the distance between Max and Kim is always non-negative. This is because the distance between two points cannot be negative.

Graphical Representation

To visualize the equation, we can plot the distance between Max and Kim over time. The graph will show a linear decrease in distance over time, with Max moving towards Kim at a constant speed.

Mathematical Representation

The equation $d=|150-9t|$ can be represented mathematically as:

d = \begin{cases} 150 - 9t & \text{if } t \leq 16.67 \\ 9t - 150 & \text{if } t > 16.67 \end{cases}

This representation shows that the distance between Max and Kim changes over time, with Max moving towards Kim at a constant speed.

Conclusion

Max rides his scooter toward Kim and then passes her at a constant speed. The distance between Max and Kim $t$ seconds after he started riding his scooter is given by $d=|150-9t|$ .

In this article, we will answer some frequently asked questions about the distance equation and provide further insights into the problem.

Q: What is the initial distance between Max and Kim?

A: The initial distance between Max and Kim is 150 units. This is the distance between them when Max starts riding his scooter.

Q: What does the 9 in the equation represent?

A: The 9 in the equation represents the constant speed at which Max moves towards Kim. This means that Max travels 9 units every second.

Q: What is the significance of the absolute value expression?

A: The absolute value expression $|150-9t|$ ensures that the distance between Max and Kim is always non-negative. This is because the distance between two points cannot be negative.

Q: How does the distance between Max and Kim change over time?

A: The distance between Max and Kim changes linearly over time. As Max moves towards Kim at a constant speed, the distance between them decreases.

Q: What happens when Max passes Kim?

A: When Max passes Kim, the distance between them becomes zero. This occurs when the expression $150-9t$ becomes zero, which happens when $t=16.67$ seconds.

Q: What is the distance between Max and Kim after Max passes Kim?

A: After Max passes Kim, the distance between them becomes negative. However, since the distance cannot be negative, we take the absolute value of the expression $9t-150$ to obtain the correct distance.

Q: Can you provide a graphical representation of the distance equation?

A: Yes, the graph of the distance equation is a linear function that decreases over time. The graph shows the distance between Max and Kim at any given time $t$ .

Q: How can we use the distance equation in real-world applications?

A: The distance equation can be used in various real-world applications, such as:

Traffic flow modeling: The distance equation can be used to model the flow of traffic on a road network.
Pedestrian movement: The distance equation can be used to model the movement of pedestrians in a crowded area.
Robotics: The distance equation can be used to model the movement of robots in a robotic system.

Q: Can you provide a mathematical representation of the distance equation?

A: Yes, the distance equation can be represented mathematically as:

d = \begin{cases} 150 - 9t & \text{if } t \leq 16.67 \\ 9t - 150 & \text{if } t > 16.67 \end{cases}

This representation shows that the distance between Max and Kim changes over time, with Max moving towards Kim at a constant speed.

Conclusion

In conclusion, the distance equation $d=|150-9t|$ represents the distance between Max and Kim at any given time $t$ . The equation shows that the distance between them changes linearly over time, with Max moving towards Kim at a constant speed. We hope that this Q&A article has provided further insights into the problem and has helped to clarify any questions you may have had.