A fence extends from the side of a house. A dog is tied to a fence post ten feet from the side of the house. The dog’s leash is six feet long. Let d be the greatest and least distance from the house that the dog can reach along the fence. Find the least and greatest distances the dog can reach.


To solve this problem, we need to calculate both the least and greatest distances from the house that the dog can reach along the fence. Here are the steps:

### Problem Setup:
- The fence post is 10 feet away from the house.
- The dog’s leash is 6 feet long.

Let \( d_{\text{min}} \) be the least distance from the house that the dog can reach, and let \( d_{\text{max}} \) be the greatest distance from the house that the dog can reach.

### 1. **Finding the Least Distance \( d_{\text{min}} \):**

The least distance occurs when the dog is directly between the house and the fence post. In this case, the dog is pulling the leash directly toward the house.

\[ d_{\text{min}} = \text{Distance from the house to the fence post} - \text{Leash length} \]

\[ d_{\text{min}} = 10 \, \text{feet} - 6 \, \text{feet} = 4 \, \text{feet} \]

### 2. **Finding the Greatest Distance \( d_{\text{max}} \):**

The greatest distance occurs when the dog is directly on the other side of the fence post, moving away from the house.

\[ d_{\text{max}} = \text{Distance from the house to the fence post} + \text{Leash length} \]

\[ d_{\text{max}} = 10 \, \text{feet} + 6 \, \text{feet} = 16 \, \text{feet} \]

### Final Answer:
- The **least distance** \( d_{\text{min}} \) from the house that the dog can reach is **4 feet**.
- The **greatest distance** \( d_{\text{max}} \) from the house that the dog can reach is **16 feet**.

Would you like more details on the geometry or the reasoning behind this problem?

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Here are five related questions to deepen your understanding:
1. How would the distances change if the leash length was different?
2. What happens to the greatest distance if the dog is tied closer to the house, say 5 feet away?
3. How can you represent the area that the dog can cover along the fence as a function of the leash length?
4. What if the dog were tied to the house instead of the fence post? What would be the reachable distances?
5. How could you calculate the maximum area the dog can roam if the leash allows movement in all directions?

**Tip:** When solving geometry-related word problems, sketching a diagram can help visualize the relationships between distances and simplify calculations.

Learn how to calculate the minimum and maximum distances a dog can reach along a fence tied to a fence post, given the distance from the house and the length of the leash. This problem involves basic geometry concepts and is suitable for students in Grades 7-9.

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