2.38. Through the median $B E$ of the base $A B C$ of the pyramid $A B C D$ and the midpoint $F$ of the edge $D C$, a plane is drawn. Find the volume of the figure $A D B F E$, if the volume of the pyramid $A B C D$ is $40 \mathrm{~cm}^{3}$.
2.38. The volume of the figure $A D B F E$ is equal to the difference in volumes of pyramids $A B C D$ and $E C B F$ (Fig. 2.33). To find the volume of pyramid $E C B F$, we compare it with the volume of pyramid $A B C D$. For this, it is sufficient to find the ratios of the areas of their bases and corresponding heights. Since the median of a triangle divides its area into two equal parts, $S_{\triangle B E C}=\frac{1}{2} S_{\triangle A B C} \cdot$ Further,

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-476.jpg?height=404&width=389&top_left_y=1473&top_left_x=767)

Fig. 2.33
since $F$ is the midpoint of edge $D C$, the height of pyramid $E C B F$ is half the height of pyramid $A B C D$. Therefore, $V_{E C B F}=\frac{1}{4} V_{A B C D}=10\left(\mathrm{~cm}^{3}\right)$. The desired volume is $30 \mathrm{~cm}^{3}$.

Answer: $30 \mathrm{~cm}^{3}$.