22. Bound the Box?

Let $C$ be an $n$-dimensional hypercube of side length $a$ centered at the point $P=(p_1,p_2,\ldots,p_n)$ in $\mathbb{R}^n$, with its edges parallel to the coordinate axes.

Derive the equations of:

1. the hypersphere inscribed in the hypercube, tangent to every $(n-1)$-dimensional face; and
2. the hypersphere circumscribed about the hypercube, passing through all $2^n$ vertices.

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