21. Bound the Box

Let $C$ be a cube of side length $a$ centered at the point $P=(p_1,p_2,p_3)$ in $\mathbb{R}^3$, with its edges parallel to the coordinate axes.

Derive the equations of:

1. the sphere inscribed in the cube, tangent to all six faces; and
2. the sphere circumscribed about the cube, passing through all eight vertices.

Express both equations in terms of $a$ and $P$, and justify your derivation.