AMC10 2018 B

AMC10 2018 B · Q10

AMC10 2018 B · Q10. It mainly tests Coordinate geometry, 3D geometry (volume).

In the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?

在所示的直角平行六面体中，$AB=3$，$BC=1$，$CG=2$。点$M$是$\overline{FG}$的中点。底面$BCHE$、顶点$M$的直角锥体积是多少？

(A) 1 1

(B) $\frac{4}{3}$ $\frac{4}{3}$

(D) $\frac{5}{3}$ $\frac{5}{3}$

(E) 2 2

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): The volume of the rectangular pyramid with base $BCHE$ and apex $M$ equals the volume of the given rectangular parallelepiped, which is $6$, minus the combined volume of triangular prism $AEHDCB$, tetrahedron $BEFM$, and tetrahedron $CGHM$. Tetrahedra $BEFM$ and $CGHM$ each have three right angles at $F$ and $G$, respectively, and the edges of the tetrahedra emanating from $F$ and $G$ have lengths $2$, $3$, and $\frac{1}{2}$, so the volume of each of these tetrahedra is $\frac{1}{6}\cdot\left(2\cdot 3\cdot \frac{1}{2}\right)=\frac{1}{2}$. The volume of the triangular prism $AEHDCB$ is $3$ because it is half the volume of the rectangular parallelepiped. Therefore the requested volume is $6-3-\frac{1}{2}-\frac{1}{2}=2$.

答案（E）：以 $BCHE$ 为底、$M$ 为顶点的长方锥体体积，等于所给长方体（长方平行六面体）的体积 $6$，减去三棱柱 $AEHDCB$、四面体 $BEFM$ 和四面体 $CGHM$ 的总体积。四面体 $BEFM$ 与 $CGHM$ 分别在 $F$ 与 $G$ 处各有三个直角，并且从 $F$ 与 $G$ 发出的三条棱长分别为 $2$、$3$ 和 $\frac{1}{2}$，因此每个四面体的体积为 $\frac{1}{6}\cdot\left(2\cdot 3\cdot \frac{1}{2}\right)=\frac{1}{2}$。三棱柱 $AEHDCB$ 的体积为 $3$，因为它是该长方体体积的一半。所以所求体积为 $6-3-\frac{1}{2}-\frac{1}{2}=2$。

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