9. [6] Compute the side length of the largest cube contained in the region
$$
\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 25 \text { and } x \geq 0\right\}
$$
of three-dimensional space.
face centered at the origin and the four vertices of the opposite face on the spherical surface. Let the side length of this cube be $s$. Then, the radius of the circle is the hypotenuse of a triangle with side lengths $s$ and $\frac{\sqrt{2}}{2} s$. So, by the Pythagorean Theorem, the radius equals $\frac{\sqrt{6}}{2} s$. Since the radius of the hemisphere is 5 , the side length of the cube is $\frac{5 \sqrt{6}}{3}$.