To determine the area of the triangle formed by the path of the cat, we need to follow these steps:
1. Identify the lengths of the sides of the triangle:
- The cat runs 120 feet due east.
- The cat then turns through an angle of [tex]\( 67.8^\circ \)[/tex] and walks 362 feet.
- We need to determine the length of the third side, which is the distance the cat walks back to the starting point.
2. Find the length of the third side:
To find the length of the third side, we use the Law of Cosines.
The Law of Cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of two sides of the triangle.
- [tex]\( C \)[/tex] is the included angle.
- [tex]\( c \)[/tex] is the length of the side opposite the angle [tex]\( C \)[/tex].
Here:
[tex]\[
a = 120 \, \text{feet}, \quad b = 362 \, \text{feet}, \quad C = 67.8^\circ
\][/tex]
Plugging in these values:
[tex]\[
c = \sqrt{120^2 + 362^2 - 2 \times 120 \times 362 \times \cos(67.8^\circ)}
\][/tex]
[tex]\[
c \approx 335.58 \, \text{feet}
\][/tex]
3. Use Heron's formula to find the area of the triangle:
- Calculate the semi-perimeter [tex]\( s \)[/tex]:
[tex]\[
s = \frac{a + b + c}{2}
\][/tex]
[tex]\[
s = \frac{120 + 362 + 335.58}{2} \approx 408.79 \, \text{feet}
\][/tex]
- Now, apply Heron's formula, which states:
[tex]\[
\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
So, substituting the values:
[tex]\[
\text{Area} = \sqrt{408.79 \times (408.79 - 120) \times (408.79 - 362) \times (408.79 - 335.58)}
\][/tex]
4. Calculate the area:
Plugging in the values:
[tex]\[
\text{Area} \approx \sqrt{408.79 \times 288.79 \times 46.79 \times 73.21} \approx 20109.91 \, \text{square feet}
\][/tex]
Therefore, the area of the triangle formed by the cat's path is:
[tex]\[
\boxed{20109.91} \, \text{square feet}
\][/tex]
