To solve this problem, let's use the Law of Sines properly applied to the given scenario. The Law of Sines states:
[tex]\[
\frac{\sin (A)}{a}=\frac{\sin (B)}{b}=\frac{\sin (C)}{c}
\][/tex]
Given the problem:
- Team A is 2.4 meters away from the chest.
- Team B is 3.2 meters away from the chest.
- The teams are 4.6 meters away from each other.
- The angle between their ropes (angle [tex]\(C\)[/tex]) is [tex]\(110^{\circ}\)[/tex].
Here [tex]\(a, b, c\)[/tex] are the sides of the triangle, and [tex]\(A, B, C\)[/tex] are the angles opposite those sides respectively. We are given:
- [tex]\(a = 2.4 \text{ meters}\)[/tex] (distance from Team A to the chest)
- [tex]\(b = 3.2 \text{ meters}\)[/tex] (distance from Team B to the chest)
- [tex]\(c = 4.6 \text{ meters}\)[/tex] (distance between the teams)
Also, we know [tex]\( \angle C = 110^{\circ} \)[/tex].
We are looking for the equation to solve for angle [tex]\(A\)[/tex]. According to the law of sines, we have:
[tex]\[
\frac{\sin (A)}{a}=\frac{\sin (C)}{c}
\][/tex]
Plugging in the known values:
[tex]\[
\frac{\sin (A)}{2.4}=\frac{\sin \left(110^{\circ}\right)}{4.6}
\][/tex]
Thus, the correct equation that can be used to solve for angle [tex]\(A\)[/tex] is:
[tex]\[
\frac{\sin (A)}{2.4}=\frac{\sin \left(110^{\circ}\right)}{4.6}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{\sin (A)}{2.4}=\frac{\sin \left(110^{\circ}\right)}{4.6}}
\][/tex]
