Question

A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is

• A. $36\pi -34$
  
• B. $30\pi -9\sqrt{3}$
• C. $30\pi -15$
• D. $35\pi -9\sqrt{3}$

Answer 1

The correct option is B $30\pi -9\sqrt{3}$

Let $O$ denote the center of the ploti and $M$ is the midpoint of the side of the walk not passing through $O$. If $AB$ denotes one arc connecting opposite sides of the walk, then $OMBA$ consists of the ${30}^{\circ }$ sector $OAB$ and the ${30}^{\circ }-{60}^{\circ }-{90}^{\circ }$ triangle $OMB$.


Thus the area of the walk is $$2[\frac{1}{12}\pi 6^2+\frac{1}{2}(3\left)3\sqrt{3}\right]=6\pi +9\sqrt{3}$$ and the required area is
$$\pi 6^2-(6\pi +9\sqrt{3})=30\pi -9\sqrt{3}squarefeet.$$ ∴ (D) is the correct option.