Solve the triangle $A B C$, if the triangle exists. \[ B=35^{\circ} 18^{\prime} \quad a=38.6 \quad b=30.9 \] Select the correct choice below and fill in the answer boxes within the choice. A. There are 2 possible solutions for the triangle. The measurements for the solution with the longer side $c$ are as follows. \[ \mathrm{m} \angle \mathrm{A}=\square^{\circ}, \quad \mathrm{m} \angle \mathrm{C}= \] The length of side $c=$ (Simplify your answer. Round to the nearest (Round to the nearest tenth degree as needed. Round to the nearest minute as needed.) as needed.) The measurements for the solution with the shorter side $\mathrm{c}$ are as follows. \[ \mathrm{m} \angle \mathrm{A}=\mathrm{O}^{\circ}, \quad \mathrm{m} \angle \mathrm{C}= \] The length of side $c=$ (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute (Round to the nearest tenth as needed.) as needed.) B. There is only 1 possible solution for the triangle. The measurements for the remaining angles $A$ and $C$ and side $C$ are as follows. $\mathrm{m} \angle \mathrm{A}=$ \[ \mathrm{m} \angle \mathrm{C}= \] (Simplify your answer. Round to the nearest degree as needed. Round to the nearest minute The length of side $\mathrm{c}=$ as needed.) (Round to the nearest tenth as needed.) C. There are no possible solutions for this triangle.
Solution
Step 1 ：Given that $B = 35^{\circ} 18^{\prime}$, $a = 38.6$, and $b = 30.9$.
Step 2 ：Convert angle $B$ to radians: $B = 0.6161012259539983$ radians.
Step 3 ：Use the Law of Cosines to find the third side $c$: $c = \sqrt{a^2 + b^2 - 2ab\cos(B)} = 22.313451136379133$.
Step 4 ：Use the Law of Sines to find the other two angles $A$ and $C$.
Step 5 ：First, find angle $A$: $A = \arcsin(\frac{a\sin(B)}{c})$. There are two possible solutions: $A1 = 88.45167567542616^{\circ}$ and $A2 = 91.54832432457384^{\circ}$.
Step 6 ：Then, find angle $C$: $C = 180^{\circ} - A - B$. There are also two possible solutions: $C1 = 90.93222309861984^{\circ}$ and $C2 = 87.83557444947216^{\circ}$.
Step 7 ：Since the sine of the angles $A1$, $A2$, $C1$, and $C2$ are all between -1 and 1, there are two possible solutions for the triangle.
Step 8 ：\boxed{\text{Final Answer:}}
Step 9 ：There are 2 possible solutions for the triangle. The measurements for the solution with the longer side $c$ are as follows: $\mathrm{m} \angle \mathrm{A}=88.45^{\circ}$, $\mathrm{m} \angle \mathrm{C}=90.93^{\circ}$, and the length of side $c=22.31$.
Step 10 ：The measurements for the solution with the shorter side $c$ are as follows: $\mathrm{m} \angle \mathrm{A}=91.55^{\circ}$, $\mathrm{m} \angle \mathrm{C}=87.84^{\circ}$, and the length of side $c=22.31$.
