9. $[\mathbf{7}]$ You are standing in an infinitely long hallway with sides given by the lines $x=0$ and $x=6$. You start at $(3,0)$ and want to get to $(3,6)$. Furthermore, at each instant you want your distance to $(3,6)$ to either decrease or stay the same. What is the area of the set of points that you could pass through on your journey from $(3,0)$ to $(3,6)$ ?
Answer: $9 \sqrt{3}+\frac{21 \pi}{2}$
If you draw concentric circles around the destination point, the condition is equivalent to the restriction that you must always go inwards towards the destination. In the diagram above, the regions through which you might pass are shaded.
We find the areas of regions A, B, and C separately, and add them up (doubling the area of region A, because there are two of them).
The hypotenuse of triangle $\mathrm{A}$ is of length 6 , and the base is of length 3 , so it is a $\frac{\pi}{6}-\frac{\pi}{3}-\frac{\pi}{2}$ triangle (30-60-90 triangle) with area $\frac{9 \sqrt{3}}{2}$. Then the total area of the regions labeled $\mathrm{A}$ is $9 \sqrt{3}$.
Since the angle of triangle A nearest the center of the circle (the destination point) is $\frac{\pi}{3}$, sector $\mathrm{B}$ has central angle $\frac{\pi}{3}$. Then the area of sector $\mathrm{B}$ is $\frac{1}{2} r^{2} \theta=\frac{1}{2} \cdot 36 \cdot \frac{\pi}{3}=6 \pi$.
Region $\mathrm{C}$ is a half-disc of radius 3 , so its area is $\frac{9 \pi}{2}$.
Thus, the total area is $9 \sqrt{3}+\frac{21 \pi}{2}$.