Answer
$3.464\text{ feet}$
Work Step by Step
Here, $r$ denotes the radius of the sector of the circle.
The Area $A$ of a sector can be computed using the formula
$A=\frac{\theta}{2}\cdot r^2$
where $\theta$ is the central angle measure in radians.
Hence, here we have
$2=\dfrac{\frac{1}{3}}{2}\cdot r^2\\
2=\frac{1}{6}\cdot r^2\\
6(2)=\frac{1}{6}r^2(6)\\
12=r^2\\
r^2=12$
Take the square root of both sides:
$\sqrt{r^2}=\pm\sqrt{12}\\
r=\pm\sqrt{12}\\
r=\pm3.4641
r\approx3.464\text{ feet}$
Since $r$ is a radius, it can't be negative.
Thus, $r\approx 3.464\text{ feet}$.