We are asked to find the area of the shaded sector of a circle. The circle has a radius of $2$ meters, and the central angle of the sector is $51$ degrees. We need to round the answer to the nearest tenth.

1. Problem Description

We are asked to find the area of the shaded sector of a circle. The circle has a radius of $2$ meters, and the central angle of the sector is $51$ degrees. We need to round the answer to the nearest tenth.

2. Solution Steps

The area of a sector of a circle is given by the formula:

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

where $\theta$ is the central angle in degrees, and $r$ is the radius of the circle. In this case, $\theta = 51$ degrees and $r = 2$ meters.

Substituting these values into the formula, we get:

$Area = \frac{51}{360} \pi (2)^2$

$Area = \frac{51}{360} \pi (4)$

$Area = \frac{51 \cdot 4 \cdot \pi}{360}$

$Area = \frac{204 \pi}{360}$

$Area = \frac{17 \pi}{30}$

Now, we can approximate $\pi$ as $3.14159$:

$Area = \frac{17 \times 3.14159}{30}$

$Area = \frac{53.40603}{30}$

$Area = 1.780201$

Rounding to the nearest tenth, we get $1.8$.

3. Final Answer

The area of the shaded sector is $1.8$ $m^2$.