1 . 某工厂欲加工一件艺术品,需要用到三棱锥形状的坯材,工人将如图所示的长方
材料切割成三棱锥
.
![](https://img.xkw.com/dksih/QBM/2016/7/5/1572883895296000/1572883900850176/STEM/1450807eaf854de487145d59f60ef73f.png)
(Ⅰ)若点
分别是棱
的中点,点
是
上的任意一点,求证:
;
(Ⅱ)已知原长方体材料中,
,
,
,根据艺术品加工需要,工程师必须求出该三棱锥的高;
(i)甲工程师先求出
所在直线与平面
所成的角
,再根据公式
求出三棱锥
的高
.请你根据甲工程师的思路,求该三棱锥的高;
(ii)乙工程师设计了一个求三棱锥的高度的程序,其框图如右图所示,则运行该程序时乙工程师应输入的
的值是多少?(请直接写出
的值,不要求写出演算或推证的过程)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6e2c5a15ee96de5510c38f68069f56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07f08e3fae189d0de5cb992e4007fca9.png)
![](https://img.xkw.com/dksih/QBM/2016/7/5/1572883895296000/1572883900850176/STEM/1450807eaf854de487145d59f60ef73f.png)
(Ⅰ)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de7878423936bda0d3fe504c4cd81bfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93a9d8b29c7c25bf005c8fa849212eb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb949f2a7ee89566c5c02b75043f6a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dffc708961cf03f696473e68986b7dd1.png)
(Ⅱ)已知原长方体材料中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0d5a2cd05e4476fc72271e8fdb59a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827064e34563a7f678020ee1fa9b1683.png)
(i)甲工程师先求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4734735213b599a9915e1ed91a5d8ce4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26ab55147ca5c0f958af43b0637ee31b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c5f8d58ee8a1fca928c91bc384d014a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07f08e3fae189d0de5cb992e4007fca9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
(ii)乙工程师设计了一个求三棱锥的高度的程序,其框图如右图所示,则运行该程序时乙工程师应输入的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://img.xkw.com/dksih/QBM/2016/7/5/1572883895296000/1572883900850176/STEM/1d4dbe7a351a4097b41c15db6f6744d6.png)
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