1973 AMC 12 Problem 25

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

36\pi-34

30\pi-15

36\pi-33

35\pi-9\sqrt3

30\pi-9\sqrt3

Bisect the path by the perpendicular diameter of the circular plot

Each half of the path is a

30^\circ

sector together with a

30^\circ

-

60^\circ

-

90^\circ

triangle

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.

Problem 25 in other years