First find the diagonal of the rectangle, since that is the circle’s radius. Using the Pythagorean theorem, the diagonal is $\sqrt{12^2+8^2}=\sqrt{144+64}=\sqrt{208}=4\sqrt{13}$. So the radius of the circle is $4\sqrt{13}$. The area of the circle is then $\pi r^2=\pi (4\sqrt{13}{)}^2=\pi (16\cdot 13)=208\pi$. Since $\pi \approx 3.14$, $208\pi \approx 653$, which is greater than 500. Therefore, the statement that must be true is that the area of the circle is greater than 500.

The water needed is the volume of the unfilled part of the cylinder. Since the container is 10 inches tall and the water is 6 inches deep, the remaining height is $10-6=4$ inches. The volume of a cylinder is $V=\pi r^{2}h$. Using $r=4$ and $h=4$, the needed volume is $V\approx 3.14(4^2)(4)=3.14(16)(4)=3.14(64)=200.96$. Therefore, $200.96$ cubic inches of water must be added.

Because the circle fits exactly inside the square, the diameter of the circle is equal to the side length of the square. The radius is 9 feet, so the diameter is $18$ feet. That means the square has side length $18$, and its area is $18^2=324$ square feet. The area of the splash pad is the area of the circle: $\pi (9^2)=81\pi$ square feet. The part of the square not covered by the splash pad is $324-81\pi$. Factoring out 81 gives $81(4-\pi )$. So the correct answer is $81(4-\pi )$ square feet.

The area of a rectangle is found by multiplying its length and width, so the rectangle's area is $(x+3)(x-1)$. The area of a circle with radius $x$ is $\pi x^2$. Since the problem states that the two areas are equal, set these expressions equal: $(x+3)(x-1)=\pi x^2$. Therefore, choice A is correct.

The area of a sector is the fraction of the full circle's area determined by its central angle. A full circle is $360^{\circ }$, so a $120^{\circ }$ sector represents $\frac{120}{360}=\frac{1}{3}$ of the circle. The area of the full circle is $\pi r^2$, so the sector's area is $\frac{1}{3}(\pi r^2)=\frac{1}{3}\pi r^2$. Therefore, the correct answer is $\frac{1}{3}\pi r^2$.