The problem asks to find the area of the shaded sector of a circle, given the radius and the central angle in degrees. We are given that the radius of the circle is $12.5$ meters and the central angle of the shaded sector is $243^\circ$. We need to round the answer to the nearest tenth.

1. Problem Description

The problem asks to find the area of the shaded sector of a circle, given the radius and the central angle in degrees. We are given that the radius of the circle is $12.5$ meters and the central angle of the shaded sector is $243^\circ$. We need to round the answer to the nearest tenth.

2. Solution Steps

The area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is given by the formula:

$A = \pi r^2 \cdot \frac{\theta}{360}$

In this problem, we have $r = 12.5$ and $\theta = 243^\circ$. Substituting these values into the formula gives:

$A = \pi (12.5)^2 \cdot \frac{243}{360}$

$A = \pi (156.25) \cdot \frac{243}{360}$

$A = \pi (156.25) \cdot 0.675$

$A = \pi \cdot 105.46875$

Using the value $\pi \approx 3.14159$, we have:

$A \approx 3.14159 \cdot 105.46875 \approx 331.355$

Rounding to the nearest tenth, we get $331.4$.

3. Final Answer

The area of the shaded sector is approximately $331.4$ $m^2$.