Answer :
To determine the area of the triangle formed by the dog's path, we use the following steps:
### Step 1: Understand the Scenario
The dog walks along two sides of a triangle. The first side [tex]\(a\)[/tex] is 180 feet, the second side [tex]\(b\)[/tex] is 329 feet, and the angle between them is [tex]\(81.5^\circ\)[/tex].
### Step 2: Use the Area Formula for a Triangle
The area [tex]\(A\)[/tex] of the triangle can be calculated using the formula:
[tex]\[ A = \frac{1}{2} \times a \times b \times \sin(C) \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the two sides, and [tex]\(C\)[/tex] is the included angle.
### Step 3: Convert the Angle to Radians
To use the sine function correctly in calculations, we need to convert the angle from degrees to radians:
[tex]\[ 81.5^\circ = 1.4224 \, \text{radians} \][/tex]
### Step 4: Calculate the Area
Substitute the values into the formula:
[tex]\[ A = \frac{1}{2} \times 180 \, \text{feet} \times 329 \, \text{feet} \times \sin(1.4224 \, \text{radians}) \][/tex]
### Step 5: Perform the Computation
Calculate the area:
[tex]\[ A \approx 29284.76 \, \text{ft}^2 \][/tex]
### Conclusion
The area of the triangle formed by the dog's path is:
[tex]\[ \boxed{29284.76 \, \text{ft}^2} \][/tex]
This result is rounded to the nearest hundredth as required.
### Step 1: Understand the Scenario
The dog walks along two sides of a triangle. The first side [tex]\(a\)[/tex] is 180 feet, the second side [tex]\(b\)[/tex] is 329 feet, and the angle between them is [tex]\(81.5^\circ\)[/tex].
### Step 2: Use the Area Formula for a Triangle
The area [tex]\(A\)[/tex] of the triangle can be calculated using the formula:
[tex]\[ A = \frac{1}{2} \times a \times b \times \sin(C) \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the two sides, and [tex]\(C\)[/tex] is the included angle.
### Step 3: Convert the Angle to Radians
To use the sine function correctly in calculations, we need to convert the angle from degrees to radians:
[tex]\[ 81.5^\circ = 1.4224 \, \text{radians} \][/tex]
### Step 4: Calculate the Area
Substitute the values into the formula:
[tex]\[ A = \frac{1}{2} \times 180 \, \text{feet} \times 329 \, \text{feet} \times \sin(1.4224 \, \text{radians}) \][/tex]
### Step 5: Perform the Computation
Calculate the area:
[tex]\[ A \approx 29284.76 \, \text{ft}^2 \][/tex]
### Conclusion
The area of the triangle formed by the dog's path is:
[tex]\[ \boxed{29284.76 \, \text{ft}^2} \][/tex]
This result is rounded to the nearest hundredth as required.
