AMC10 2015 B

AMC10 2015 B · Q17

AMC10 2015 B · Q17. It mainly tests 3D geometry (volume), Geometry misc.

The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of the octahedron?

将下图所示直角长方体各面的中心点连接起来形成一个八面体。这个八面体的体积是多少？

(A) $\frac{75}{12}$ $\frac{75}{12}$

(B) 10 10

(D) $10\sqrt{2}$ $10\sqrt{2}$

(E) 15 15

Answer

Correct choice: (B)

正确答案：（B）

Solution

Consider the octahedron to be two pyramids whose base is a rhombus in the middle horizontal plane [...]. The area of the base is [...] 10. The altitude of each pyramid is half that of the prism or 3/2. The volume of the octahedron is $2 \cdot \frac{1}{3} \cdot 10 \cdot \frac{3}{2} = 10$.

将八面体视为两个底面为中间水平面上的菱形金字塔。[...] 底面面积为 [...] 10。每个金字塔的高度为长方体高度的一半，即 3/2。八面体的体积为 $2 \cdot \frac{1}{3} \cdot 10 \cdot \frac{3}{2} = 10$。

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