jmc

Let $F$ be the center of the square base. Since the pyramid is a right pyramid, triangle $PFB$ is a right triangle. The area of the base of the pyramid is 288 square cm, so the length of each side of the base is $12\sqrt{2}$ cm. Since $F$ is the center of the base, $FB$ is half the diagonal of the base, or $(12\sqrt{2}\cdot \sqrt{2})/2=12$ cm. Applying the Pythagorean Theorem to triangle $PFB$ gives $$PF=\sqrt{P{B}^2-F{B}^2}=\sqrt{225-144}=\sqrt{81}=9,$$so the volume of the pyramid is $[ABCD](PF)/3=(288)(9)/3=288\cdot 3=\boxed{864}$ cubic centimeters.