Morgan's Dog Walking Problem Solving Minimum And Maximum Distances
Morgan's dog walking scenario presents an intriguing mathematical puzzle, allowing us to explore the concepts of absolute value and distance in a practical context. In this article, we will delve into the problem, dissect the equation provided, and determine the minimum and maximum distances Morgan's dog can be from her house while adhering to the leash length constraint. Understanding this problem not only reinforces mathematical principles but also highlights how math can be applied to everyday situations. So, let's embark on this journey of calculations and explore the boundaries of Morgan's dog walking adventure.
The core of the problem lies in understanding the equation |x - 500| = 8. This equation encapsulates the relationship between the dog's distance from the house (represented by x), Morgan's distance from the house (500 meters), and the leash length (8 meters). The absolute value notation, denoted by the vertical bars, signifies that we are interested in the magnitude of the difference between the dog's distance and Morgan's distance, irrespective of the direction. This is crucial because the dog could be either closer to the house than Morgan or farther away, but the leash length remains the same in both scenarios.
To effectively solve this equation, we need to consider two possibilities: first, where the expression inside the absolute value is positive or zero, and second, where it is negative. When (x - 500) is positive or zero, the absolute value simply removes the parentheses, and the equation becomes x - 500 = 8. On the other hand, when (x - 500) is negative, the absolute value negates the expression, resulting in -(x - 500) = 8. By addressing these two cases separately, we can unveil the two possible distances the dog can be from the house, corresponding to the maximum and minimum distances allowed by the leash length.
To determine the minimum and maximum distances, we must solve the absolute value equation |x - 500| = 8. This equation implies that the distance between the dog's position (x) and Morgan's position (500 meters from home) is exactly 8 meters. The absolute value means we need to consider two separate cases:
- Case 1: x - 500 = 8
- Case 2: -(x - 500) = 8
In Case 1, we have a straightforward linear equation: x - 500 = 8. To solve for x, we add 500 to both sides of the equation: x = 500 + 8, which simplifies to x = 508. This solution represents the maximum distance the dog can be from the house. When the dog is at this distance, it is 8 meters farther away from the house than Morgan is.
Turning our attention to Case 2, we have -(x - 500) = 8. To simplify this, we first distribute the negative sign: -x + 500 = 8. Next, we isolate x by subtracting 500 from both sides: -x = 8 - 500, which simplifies to -x = -492. Finally, we multiply both sides by -1 to solve for x: x = 492. This solution represents the minimum distance the dog can be from the house. In this scenario, the dog is 8 meters closer to the house than Morgan is.
By solving these two cases, we have found the two critical distances. The maximum distance the dog can be from the house is 508 meters, and the minimum distance is 492 meters. These values define the boundaries within which the dog can roam while remaining within the constraints of the 8-meter leash.
After solving the two equations derived from the absolute value equation, we have obtained two possible distances: 492 meters and 508 meters. These distances represent the boundaries within which the dog can move while staying within the 8-meter leash length from Morgan, who is 500 meters from her house. To determine which distance is the minimum and which is the maximum, we simply compare the two values.
It is evident that 492 meters is less than 508 meters. Therefore, 492 meters represents the minimum distance the dog can be from the house. This scenario occurs when the dog is on the side of Morgan closer to the house, stretching the leash to its full 8-meter length in that direction. In this case, the dog is 492 meters away from the house, while Morgan remains 500 meters away.
Conversely, 508 meters is the maximum distance the dog can be from the house. This occurs when the dog is on the side of Morgan farther from the house, again stretching the leash to its full 8-meter length. At this point, the dog is 508 meters away from the house, marking the farthest it can venture while staying connected to Morgan by the leash.
In summary, the minimum distance the dog can be from the house is 492 meters, and the maximum distance is 508 meters. These values provide a clear range of the dog's possible positions relative to the house, given the leash length and Morgan's location.
This exercise of calculating minimum and maximum distances has practical implications beyond the realm of mathematical equations. It illustrates how mathematical concepts can be applied to real-world scenarios, such as understanding spatial relationships and constraints. In Morgan's case, knowing the minimum and maximum distances her dog can be from the house can help her ensure the dog's safety and prevent it from wandering too far.
Furthermore, this problem highlights the utility of absolute value equations in representing distances and ranges. The absolute value notation allows us to express the distance between two points without considering direction, which is crucial in many real-world applications. Whether it's calculating distances on a map, determining the tolerance in manufacturing processes, or, as in this case, managing a dog's leash length, absolute value equations provide a powerful tool for problem-solving.
In conclusion, by carefully analyzing the equation |x - 500| = 8 and considering the two possible cases, we have successfully determined that the minimum distance the dog can be from the house is 492 meters, and the maximum distance is 508 meters. This exercise not only reinforces our understanding of absolute value equations but also demonstrates the practical relevance of mathematics in everyday life. Morgan can now confidently walk her dog, knowing the boundaries within which her furry companion can safely roam.
