Find the unknown angles in triangle $\mathrm{ABC}$ for each triangle that exists.
\[
A=83.6^{\circ} \quad b=9.1 \mathrm{ft} \quad a=12.2 \mathrm{ft}
\]
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round to the nearest tenth as needed.)
A. There are two possible solutions for the triangle. The measurements for when $B$ is larger are $B=$
B. There is only one possible solution for the triangle. The measurements for the remaining angles are B
C. There are no possible solutions for the triangle.

Step 1 ：We are given one angle and two sides of a triangle. This is a case of solving a triangle using the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

Step 2 ：We can use the Law of Sines to find angle B. The formula for the Law of Sines is: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). We can rearrange this formula to solve for B: \(B = \sin^{-1}\left(\frac{b \sin A}{a}\right)\).

Step 3 ：We can plug in the given values into this formula to find B. However, we need to be aware that the sine function has two possible values for each input in the range of 0 to 180 degrees, one in the first quadrant (0 to 90 degrees) and one in the second quadrant (90 to 180 degrees). Therefore, there may be two possible values for B, one acute and one obtuse.

Step 4 ：After finding B, we can find angle C by subtracting the sum of angles A and B from 180 degrees, because the sum of the angles in a triangle is 180 degrees.

Step 5 ：We have two possible solutions for the triangle. The first solution is when angle B is acute (47.8 degrees) and angle C is obtuse (130.7 degrees). The second solution is when angle B is obtuse (132.2 degrees) and angle C is acute (46.4 degrees). However, we need to check if these solutions are valid. A triangle is valid if the sum of its angles is 180 degrees and each angle is greater than 0 and less than 180 degrees.

Step 6 ：Both solutions are not valid. This means that there are no possible solutions for the triangle with the given values of A, a, and b.

Step 7 ：Final Answer: \(\boxed{\text{The correct choice is C. There are no possible solutions for the triangle.}}\)