Solution:

The answer is (B). Let us unroll onto the plane the lateral surface of the mountain, cutting it along the segment $DC$. We obtain a circular sector with center $C$, radius $4~\mathrm{km}$, that is, the length of $DC$, and bounded by an arc of circumference of $4\pi~\mathrm{km}$, that is, the circumference of the base of the original mountain. The unrolled figure will then be a semicircle. In particular, the point $D$ will lie at one of the endpoints of the arc bounding the semicircle, and $P$ at the midpoint of this arc, so the angle $P\widehat{C}D$ will be a right angle. Moreover $CD=4~\mathrm{km}$, the distance between $C$ and the Gate of Hell is $3~\mathrm{km}$, so the distance between $D$ and the Gate of Hell is $5~\mathrm{km}$ by the Pythagorean theorem.