A triangular prism $A B C A 1 B 1 C 1$ with the lower base $A B C$ and lateral edges $A A 1, B B 1, C C 1$ is cut by a plane passing through points $E, F, C$, where point $E$ is the midpoint of edge $A A 1$, and point $F$ lies on edge $B B 1$ with $B F: F B 1=1: 2$. Find the volume of the part of the prism $A B C A 1 B 1 C 1$ enclosed between the cutting plane and the lower base of the prism, given that the volume of the prism is $V$.
Let the lateral edges of the prism be $6a$, the distance between the lateral edges $AA1$ and $BB1$ be $h$, and the distance from vertex $C$ to the plane of the face $AA1B1B$ be $H$. Construct triangles $ABC$ and $A1B1C1$ into parallelograms $ABDC$ and $A1B1D1C1$. We obtain a parallelepiped $ABDC A1B1D1C1$ with base $AA1B1B$ and lateral edges $A1C1, AC, BD$, and $B1D1$. Let $V1$ be its volume. Then

$$
V1 = S_{AA1B1B} \cdot H = 6ah \cdot H
$$

Therefore,

$$
V = \frac{1}{2} V1 = 3ah \cdot H
$$

Let $V2$ be the volume of the quadrilateral pyramid $CABFE$ with vertex $C$. The base of this pyramid is the trapezoid $ABFE$. Then

## Answer

$\frac{5}{18} V$.