We are given a triangle $ABC$ with $C = 96.2^{\circ}$, $b = 11.2$ cm, and $c = 39.4$ cm. We are asked to find the other angles and side length. We will use the Law of Sines.

1. Problem Description

We are given a triangle $ABC$ with $C = 96.2^{\circ}$, $b = 11.2$ cm, and $c = 39.4$ cm. We are asked to find the other angles and side length. We will use the Law of Sines.

2. Solution Steps

The Law of Sines states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

We can use the Law of Sines to find $\sin B$:

$\frac{b}{\sin B} = \frac{c}{\sin C}$

$\sin B = \frac{b \sin C}{c}$

$\sin B = \frac{11.2 \sin(96.2^{\circ})}{39.4}$

$\sin B = \frac{11.2 \times 0.9940}{39.4}$

$\sin B = \frac{11.1328}{39.4} = 0.28256$

$B = \arcsin(0.28256) \approx 16.39^{\circ}$

Now we can find angle $A$ using the fact that the sum of the angles in a triangle is $180^{\circ}$:

$A + B + C = 180^{\circ}$

$A = 180^{\circ} - B - C$

$A = 180^{\circ} - 16.39^{\circ} - 96.2^{\circ}$

$A = 180^{\circ} - 112.59^{\circ}$

$A = 67.41^{\circ}$

Now we can find side $a$ using the Law of Sines:

$\frac{a}{\sin A} = \frac{c}{\sin C}$

$a = \frac{c \sin A}{\sin C}$

$a = \frac{39.4 \sin(67.41^{\circ})}{\sin(96.2^{\circ})}$

$a = \frac{39.4 \times 0.9233}{0.9940}$

$a = \frac{36.378}{0.9940}$

$a \approx 36.60$ cm

3. Final Answer

$A = 67.41^{\circ}$

$B = 16.39^{\circ}$

$a = 36.60$ cm