Morgan Is Walking Her Dog On An 8-meter-long Leash. She Is Currently 500 Meters From Her House, So The Maximum And Minimum Distances That The Dog May Be From The House Can Be Found Using The Equation ∣ X − 500 ∣ = 8 |x-500|=8 ∣ X − 500∣ = 8 . What Are The Minimum And

Introduction

In this article, we will explore the problem of finding the minimum and maximum distances that a dog may be from its house when its owner is walking it on an 8-meter-long leash. The owner, Morgan, is currently 500 meters from her house, and we need to use the equation $|x-500|=8$ to find the possible distances of the dog from the house.

The Equation: Absolute Value

The equation $|x-500|=8$ represents the absolute value of the difference between the distance of the dog from the house and 500 meters. 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, we have $|x-500|=8$, which means that the distance of the dog from the house is either 8 meters more or less than 500 meters.

Solving the Equation

To solve the equation $|x-500|=8$, we need to consider two cases:

Case 1: $x-500 \geq 0$

In this case, we have:

$$x-500 = 8$$

Solving for $x$, we get:

$$x = 508$$

This means that the dog is 508 meters from the house.

Case 2: $x-500 < 0$

In this case, we have:

$$-(x-500) = 8$$

Simplifying, we get:

$$-x+500 = 8$$

Solving for $x$, we get:

$$-x = -492$$

$$x = 492$$

This means that the dog is 492 meters from the house.

Conclusion

In conclusion, the minimum and maximum distances that the dog may be from the house are 492 meters and 508 meters, respectively. These distances are found using the equation $|x-500|=8$, which represents the absolute value of the difference between the distance of the dog from the house and 500 meters.

Minimum and Maximum Distances

The minimum distance of the dog from the house is 492 meters, which occurs when the dog is 8 meters less than 500 meters from the house. The maximum distance of the dog from the house is 508 meters, which occurs when the dog is 8 meters more than 500 meters from the house.

Graphical Representation

The graph of the equation $|x-500|=8$ is a V-shaped graph with its vertex at (500, 0). The graph has two branches, one with a positive slope and the other with a negative slope. The minimum distance of the dog from the house occurs at the point where the graph intersects the x-axis, which is at x = 492. The maximum distance of the dog from the house occurs at the point where the graph intersects the x-axis, which is at x = 508.

Real-World Applications

The problem of finding the minimum and maximum distances of a dog from its house has real-world applications in various fields, such as:

• Animal behavior: Understanding the movement patterns of animals, including dogs, is crucial in animal behavior studies.
• Wildlife conservation: Knowing the minimum and maximum distances of animals from their habitats can help conservationists develop effective conservation strategies.
• Search and rescue: In search and rescue operations, understanding the movement patterns of animals can help rescue teams locate missing persons or animals.

Conclusion

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Q: What is the minimum distance of the dog from the house?

A: The minimum distance of the dog from the house is 492 meters. This occurs when the dog is 8 meters less than 500 meters from the house.

Q: What is the maximum distance of the dog from the house?

A: The maximum distance of the dog from the house is 508 meters. This occurs when the dog is 8 meters more than 500 meters from the house.

Q: How do I calculate the minimum and maximum distances of the dog from the house?

A: To calculate the minimum and maximum distances of the dog from the house, you can use the equation $|x-500|=8$. This equation represents the absolute value of the difference between the distance of the dog from the house and 500 meters.

Q: What is the significance of the absolute value function in this problem?

A: The absolute value function is used to represent the distance of the dog from the house, which can be either positive or negative. The absolute value function ensures that the distance is always non-negative, which is essential in this problem.

Q: Can I use the equation $|x-500|=8$ to find the distance of the dog from the house at any point in time?

A: Yes, you can use the equation $|x-500|=8$ to find the distance of the dog from the house at any point in time. However, you need to consider the constraints of the problem, such as the length of the leash and the position of the owner.

Q: How does the length of the leash affect the minimum and maximum distances of the dog from the house?

A: The length of the leash affects the minimum and maximum distances of the dog from the house. A longer leash allows the dog to move further away from the house, while a shorter leash restricts the dog's movement.

Q: Can I use the equation $|x-500|=8$ to find the distance of the dog from the house in a 3D space?

A: No, the equation $|x-500|=8$ is only applicable in a 2D space. To find the distance of the dog from the house in a 3D space, you need to use a more complex equation that takes into account the x, y, and z coordinates.

Q: How does the position of the owner affect the minimum and maximum distances of the dog from the house?

A: The position of the owner affects the minimum and maximum distances of the dog from the house. The owner's position determines the starting point of the leash, which in turn affects the distance of the dog from the house.

Q: Can I use the equation $|x-500|=8$ to find the distance of the dog from the house in a real-world scenario?

A: Yes, you can use the equation $|x-500|=8$ to find the distance of the dog from the house in a real-world scenario. However, you need to consider the constraints of the problem, such as the length of the leash, the position of the owner, and the environment.

Q: What are the real-world applications of the equation $|x-500|=8$?

A: The equation $|x-500|=8$ has real-world applications in various fields, such as animal behavior, wildlife conservation, and search and rescue. Understanding the movement patterns of animals, including dogs, is crucial in these fields.

Conclusion

In conclusion, the equation $|x-500|=8$ is a powerful tool for finding the minimum and maximum distances of a dog from its house. By understanding the equation and its applications, you can gain insights into the movement patterns of animals and develop effective strategies for various fields.