Find the unknown angles in triangle $A B C$ for the following triangle that exists.
\[
C=47^{\circ} 40^{\prime}, b=22.6 m, c=32.2 m
\]
Select the correct choice below, and, if necessary, fill in the answer boxes to complete your choice.
(Do not round until the final answers. Then round to the nearest whole number as needed.)
A. There is only one possible solution for the triangle. The measurements for the remaining angles are $A=\square^{\circ} \square^{\prime}$ and
\[
\mathrm{B}=
\]
B. There are two possible solutions for the triangle. The measurements for when $B$ is larger are $A=\square^{\circ} \square^{\prime}$ and $B=\square^{\circ} \square^{\prime}$. The measurements for when $\mathrm{B}$ is smaller are $\mathrm{A}=\square^{\circ} \square^{\prime}$ and $\mathrm{B}=\square^{\circ} \square$ '
C. There are no possible solutions for this triangle.

Step 1 ：Given that we know the angle C and the sides b and c, we can first find the angle B using the formula: \(B = \sin^{-1}\left(\frac{b \sin C}{c}\right)\)

Step 2 ：Then, we can find the angle A using the fact that the sum of the angles in a triangle is 180 degrees: \(A = 180 - B - C\)

Step 3 ：We need to consider the ambiguous case of the Law of Sines, where two different triangles satisfy the given conditions. This occurs when the angle B is acute (less than 90 degrees) and the side a (opposite to angle A) is less than the height of the triangle (which is b sin B). In this case, there is another possible angle for B, which is: \(B' = 180 - B\) And the corresponding angle A' is: \(A' = 180 - B' - C\)

Step 4 ：We need to calculate these angles and check if they satisfy the conditions for the ambiguous case.

Step 5 ：The calculations have been done and it seems that there is no ambiguous case for this triangle, as the calculated angles A' and B' are the same as A and B, respectively.

Step 6 ：Therefore, there is only one possible solution for the triangle. The measurements for the remaining angles are A = 101° 5' and B = 31° 15'.

Step 7 ：Final Answer: \(\boxed{\text{The correct choice is A. There is only one possible solution for the triangle. The measurements for the remaining angles are } A=101^{\circ} 5^{\prime} \text{ and } B=31^{\circ} 15^{\prime}}\)