6. Dante's house is located at point $D$ at the foot of a conical mountain with a base diameter of $4 \mathrm{~km}$ and the summit at point $C$. It is known that $D$ is $4 \mathrm{~km}$ away from $C$ in a straight line and that, if $P$ is the point diametrically opposite to $D$ on the base of the mountain, the gate of Hell is located at $3 / 4$ of the segment $C P$, closer to $P$. What is the minimum distance Dante must travel (walking on the slopes of the mountain) to reach the gate of Hell from his house?
(A) $\pi+1 \mathrm{~km}$
(B) $5 \mathrm{~km}$
(C) $2 \pi \mathrm{~km}$
(D) $7 \mathrm{~km}$
(E) $2 \pi+1 \mathrm{~km}$.
6. The answer is (B). Unfolding the lateral surface of the mountain on the plane, cutting it along the segment $D C$. We will have a circular sector with center $C$, radius $4 \text{ km}$, i.e., the length of $D C$, and bounded by an arc of circumference of $4 \pi \text{ km}$, i.e., the base circumference of the original mountain. The development will then be a semicircle. In particular, the point $D$ will be at one of the ends of the arc of the circumference that delimits the semicircle, and $P$ will be at the midpoint of this arc, so the angle $P \widehat{C} D$ will be a right angle. Furthermore, $C D=4 \text{ km}$, the distance between $C$ and the gate of Hell is $3 \text{ km}$, so the distance between $D$ and the gate of Hell is $5 \text{ km}$ by the Pythagorean theorem.