To find the length of side \( x \) in the triangle, we can use the Law of Sines. The Law of Sines states that the ratio of the lengths of the sides of a triangle to the sine of the opposite angles is constant:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
In this triangle, we have:
- Side \( a = x \) (opposite angle \( A \))
- Side \( b = 14 \) cm (opposite angle \( B \))
- Angle \( A = 41^\circ \)
- Angle \( B = 62^\circ \)
First, let's calculate angle \( C \):
\[ C = 180^\circ - A - B = 180^\circ - 41^\circ - 62^\circ = 77^\circ \]
Next, we can set up the ratio using the Law of Sines:
\[ \frac{x}{\sin(41^\circ)} = \frac{14}{\sin(62^\circ)} \]
Now, we can solve for \( x \):
\[ x = \frac{14 \cdot \sin(41^\circ)}{\sin(62^\circ)} \]
Before substituting values, we will calculate \( \sin(41^\circ) \) and \( \sin(62^\circ) \):
- \( \sin(41^\circ) \approx 0.6561 \)
- \( \sin(62^\circ) \approx 0.8746 \)
Substituting these values in gives:
\[ x = \frac{14 \cdot 0.6561}{0.8746} \approx \frac{9.2254}{0.8746} \approx 10.54 \text{ cm} \]
So, rounding to one decimal place, the length of the side labeled \( x \) cm is approximately 10.5 cm. However, it seems the answer options do not include this value.
The closest answer from the options given is 10.4 cm. Therefore, the length of the side labeled \( x \) cm is:
10.4 cm.