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 $|x-500|=8$. 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 that the dog may be from the house is 492 meters, and the maximum distance is 508 meters.

Graphical Representation

The graph of the equation $|x-500|=8$ is a V-shaped graph, with the minimum and maximum distances represented by the points (492, 0) and (508, 0), respectively.

Real-World Applications

This problem has real-world applications in various fields, such as:

• Navigation: In navigation, it is essential to know the minimum and maximum distances that a vehicle or a person may be from a reference point, such as a house or a landmark.
• Surveying: In surveying, it is necessary to determine the minimum and maximum distances between two points, such as the distance between two buildings or the distance between a building and a road.
• Geography: In geography, it is essential to understand the minimum and maximum distances between two locations, such as the distance between two cities or the distance between a city and a natural feature.

Conclusion

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Q: What is the equation used to find the minimum and maximum distances of a dog from the house?

A: The equation used to find the minimum and maximum distances of a dog from the house is $|x-500|=8$, where $x$ represents the distance of the dog from the house.

Q: What is the meaning of the absolute value function in this equation?

A: The absolute value function in this equation represents the distance of the dog from the house, which can be either 8 meters more or less than 500 meters.

Q: How do we solve the equation $|x-500|=8$?

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

• Case 1: $x-500 \geq 0$, where we have $x-500 = 8$ and $x = 508$.
• Case 2: $x-500 < 0$, where we have $-(x-500) = 8$ and $x = 492$.

Q: What are the minimum and maximum distances that the dog may be from the house?

A: The minimum distance that the dog may be from the house is 492 meters, and the maximum distance is 508 meters.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields, such as:

• Navigation: In navigation, it is essential to know the minimum and maximum distances that a vehicle or a person may be from a reference point, such as a house or a landmark.
• Surveying: In surveying, it is necessary to determine the minimum and maximum distances between two points, such as the distance between two buildings or the distance between a building and a road.
• Geography: In geography, it is essential to understand the minimum and maximum distances between two locations, such as the distance between two cities or the distance between a city and a natural feature.

Q: Can this problem be solved using other methods?

A: Yes, this problem can be solved using other methods, such as:

• Graphical method: By graphing the equation $|x-500|=8$, we can visualize the minimum and maximum distances that the dog may be from the house.
• Algebraic method: By using algebraic manipulations, we can solve the equation $|x-500|=8$ and find the minimum and maximum distances.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not considering both cases of the absolute value function.
• Not solving the equation correctly.
• Not understanding the real-world applications of the problem.

Q: How can I apply this problem to real-world situations?

A: You can apply this problem to real-world situations by:

• Using the equation $|x-500|=8$ to find the minimum and maximum distances between two points.
• Understanding the real-world applications of the problem, such as navigation, surveying, and geography.
• Using the problem as a starting point to explore other mathematical concepts, such as graphing and algebraic manipulations.