Answer: E

Small Hint:

Bisect the path by the perpendicular diameter of the circular plot

Big Hint:

Each half of the path is a $30^\circ$ sector together with a $30^\circ$-$60^\circ$-$90^\circ$ triangle

Solution:

The plot has radius $6.$ At the path’s other edge, the perpendicular distance from the center is $3,$ so the radius to an intersection point makes a $30^\circ$ angle with the edge through the center. Half of the path consists of a $30^\circ$ sector of radius $6$ and a right triangle with legs $3$ and $3\sqrt3.$ Its area is $$\begin{aligned} &\frac{30}{360}\pi(6^2) +\frac12(3)(3\sqrt3)\\ &\qquad=3\pi+\frac{9\sqrt3}{2}. \end{aligned}$$ Thus the whole path has area $6\pi+9\sqrt3,$ and the remaining grass area is $$36\pi-(6\pi+9\sqrt3) =30\pi-9\sqrt3.$$

Therefore, the correct answer is E.