We are given a right triangle $ABC$ with a right angle at vertex $A$. We are given the length of side $c$, which is $42$ meters, and the angle at vertex $C$, which is $31^\circ$. We need to find the length of side $a$.

1. Problem Description

We are given a right triangle $ABC$ with a right angle at vertex $A$. We are given the length of side $c$, which is $42$ meters, and the angle at vertex $C$, which is $31^\circ$. We need to find the length of side $a$.

2. Solution Steps

We are given angle $C = 31^\circ$ and side $c = 42$m. We need to find side $a$, which is the hypotenuse of the triangle.

We can use the cosine function to relate the given angle and the side adjacent to it with the hypotenuse:

$\cos(C) = \frac{c}{a}$

Plugging in the given values:

$\cos(31^\circ) = \frac{42}{a}$

To solve for $a$, we can rearrange the equation:

$a = \frac{42}{\cos(31^\circ)}$

Using a calculator, we find that $\cos(31^\circ) \approx 0.8572$.

Therefore,

$a = \frac{42}{0.8572} \approx 48.9967$

Rounding to two decimal places, we get $a \approx 49.00$.

3. Final Answer

The length of side $a$ is approximately $49.00$ meters.