We are given a triangle $ABC$ with angle $C = 47.8^\circ$, side $a = 13.1$ m, and side $b = 24.2$ m. We need to find the length of side $c$.

1. Problem Description

We are given a triangle $ABC$ with angle $C = 47.8^\circ$, side $a = 13.1$ m, and side $b = 24.2$ m. We need to find the length of side $c$.

2. Solution Steps

We can use the Law of Cosines to find the length of side $c$. The Law of Cosines states:

$c^2 = a^2 + b^2 - 2ab \cos{C}$

Plugging in the given values, we have:

$c^2 = (13.1)^2 + (24.2)^2 - 2(13.1)(24.2) \cos{(47.8^\circ)}$

$c^2 = 171.61 + 585.64 - 634.84 \cos{(47.8^\circ)}$

We need to find the value of $\cos{(47.8^\circ)}$. Using a calculator, we find that $\cos{(47.8^\circ)} \approx 0.6713$.

Substituting this value, we have:

$c^2 = 171.61 + 585.64 - 634.84(0.6713)$

$c^2 = 757.25 - 425.14$

$c^2 = 332.11$

Now, we take the square root of both sides to find $c$:

$c = \sqrt{332.11}$

$c \approx 18.22$

3. Final Answer

The length of side $c$ is approximately 18.22 m.