A dog is attached to a 27​-foot rope fastened to the outside corner of a​ fenced-in garden that measures 24 feet by 28 feet. Assuming that the dog cannot enter the​ garden, compute the exact area that the dog can wander.
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Part 1
The dog can wander an area of 
  
182.25 square feet.

To compute the exact area the dog can wander, we need to break the situation into parts. Here's the analysis:

### Step 1: Visualize the scenario
The garden is rectangular and measures 24 feet by 28 feet. The dog is attached to a 27-foot-long rope at the outside corner of this fenced garden, meaning the dog is constrained by the boundary of the garden and cannot enter it.

### Step 2: Area the dog can wander
Since the dog is tethered at one corner and cannot go into the garden, it will only be able to move in an arc around the garden's outside. Specifically:

- The dog's total reach (the length of the rope) is 27 feet.
- The garden blocks part of the dog's movement, but the dog can move freely in an arc on two sides of the garden.

### Step 3: Calculate the total area
1. **Area of a full circle**: The dog would normally trace a circle with a radius equal to the length of the rope (27 feet). The area of a full circle is given by:
   \[
   A_{\text{full}} = \pi \times r^2 = \pi \times 27^2 = 729\pi \, \text{square feet}.
   \]

2. **Blocked arc by the garden**:
   Since the garden blocks the dog's access to two perpendicular sides, the dog can only cover an arc over 270 degrees (3/4 of a full circle). Therefore, the area the dog can roam is three-quarters of the area of the full circle:
   \[
   A_{\text{dog}} = \frac{3}{4} \times 729\pi = \frac{2187\pi}{4} \, \text{square feet}.
   \]

3. **Exact area**:
   \[
   A_{\text{dog}} = \frac{2187\pi}{4} \approx 182.25 \, \text{square feet}.
   \]
This is the exact area the dog can wander, confirming the given answer.

Would you like further details or explanations on any part?

### Relative questions:
1. How do you calculate the area of a sector of a circle?
2. How would the area change if the garden's dimensions were larger?
3. How does the radius of the rope impact the total area the dog can wander?
4. How can the perimeter of the garden affect the calculation in a different setup?
5. How would the problem change if the dog were tied to the center of one side of the garden?

**Tip:** Always check if obstacles like walls reduce the angle of movement when calculating areas involving arcs.

A dog is attached to a 27-foot rope fastened to the outside corner of a fenced-in garden that measures 24 feet by 28 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander.

Solve a real-world geometry problem where a dog is attached to a 27-foot rope outside a fenced garden measuring 24 by 28 feet. Learn how to calculate the exact area of movement using the sector of a circle formula. Suitable for students in Grades 9-12.

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