Solve the triangle $A B C$, if the triangle exists.
\[
A=43.5^{\circ} \quad a=8.8 \mathrm{~m} \quad b=10.5 \mathrm{~m}
\]
Select the correct choice below and fill in the answer boxes within the choice.
A. There is only 1 possible solution for the triangle.
The measurements for the remaining angles $B$ and $C$ and side $C$ are as follows.
\[
\mathrm{m} \angle \mathrm{B}=
\]
\[
\mathrm{m} \angle \mathrm{C}=\mathrm{O}^{\circ}
\]
(Round to the nearest
(Round to the nearest tenth as needed.)
The length of side $\mathrm{c}=$
(Round to the nearest tenth as needed.)
B. There are 2 possible solutions for the triangle.
The measurements for the solution with the longer side $c$ are as follows.
$\mathrm{m} \angle \mathrm{B}=$
$\mathrm{m} \angle \mathrm{C}=$
The length of side $c=$
(Round to the nearest
(Round to the nearest
(Round to the nearest tenth
tenth as needed.) tenth as needed.) as needed.)
The measurements for the solution with the shorter side $c$ are as follows.
$\mathrm{m} \angle \mathrm{B}=$
$\mathrm{m} \angle \mathrm{C}=$
The length of side $c=$
(Round to the nearest
(Round to the nearest
(Round to the nearest tenth tenth as needed.) tenth as needed.) as needed.)
C. There are no possible solutions for this triangle.
Answer
Final Answer: The measurements for the remaining angles $B$ and $C$ and side $C$ are as follows. \(\mathrm{m} \angle \mathrm{B}= \boxed{55.2}^{\circ}\), \(\mathrm{m} \angle \mathrm{C}= \boxed{81.3}^{\circ}\), The length of side $\mathrm{c}= \boxed{12.6} \mathrm{~m}$
Step 1 ：Given that $A=43.5^{\circ}$, $a=8.8$ m, and $b=10.5$ m, we can use the Law of Sines to find the other angles and sides.
Step 2 ：First, calculate angle B using the formula: $B = \sin^{-1}\left(\frac{b \cdot \sin(A)}{a}\right)$
Step 3 ：Substitute the given values into the formula to get $B = \sin^{-1}\left(\frac{10.5 \cdot \sin(43.5)}{8.8}\right)$, which gives $B = 55.2^{\circ}$
Step 4 ：Next, calculate angle C using the formula: $C = 180 - A - B$
Step 5 ：Substitute the given values into the formula to get $C = 180 - 43.5 - 55.2$, which gives $C = 81.3^{\circ}$
Step 6 ：Finally, calculate side c using the formula: $c = a \cdot \sin(C) / \sin(A)$
Step 7 ：Substitute the given values into the formula to get $c = 8.8 \cdot \sin(81.3) / \sin(43.5)$, which gives $c = 12.6$ m
Step 8 ：Final Answer: The measurements for the remaining angles $B$ and $C$ and side $C$ are as follows. \(\mathrm{m} \angle \mathrm{B}= \boxed{55.2}^{\circ}\), \(\mathrm{m} \angle \mathrm{C}= \boxed{81.3}^{\circ}\), The length of side $\mathrm{c}= \boxed{12.6} \mathrm{~m}$
