如圖,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342075372.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342090368.png)
為圓柱
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342106355.png)
的母線,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342121395.png)
是底面圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342137292.png)
的直徑,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342153315.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342168318.png)
分別是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342075372.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342184424.png)
的中點,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342246684.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240213422627194.png)
(1)證明:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342277713.png)
;
(2)證明:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342293728.png)
;
(3)求四棱錐
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342309614.png)
與圓柱
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342106355.png)
的體積比.
(1)詳見解析; (2) 詳見解析; (3)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342340485.png)
.
試題分析:(1)證明線面平行,可證線線平行,所以通過證明四邊形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342355539.png)
是平行四邊形可知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342371948.png)
,從而證得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342387722.png)
.(2)證明面面垂直,可證線面垂直,所以通過證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342402669.png)
,而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342418663.png)
,從而證得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342293728.png)
.(3)關鍵是求四棱錐的高,通過證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342449680.png)
找到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342465396.png)
就是棱錐的高,再分別利用圓柱和棱錐的體積公式計算.
試題解析:(1)證明:連結
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342480395.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342511376.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342527455.png)
分別為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342543516.png)
的中點,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342558581.png)
.
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342574565.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342589843.png)
.∴四邊形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342355539.png)
是平行四邊形,
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342371948.png)
. ∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342387722.png)
. 4分
(2) 證明:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342075372.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342090368.png)
為圓柱
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342106355.png)
的母線,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342714590.png)
且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342730546.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342745533.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342121395.png)
是底面圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342137292.png)
的直徑,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342792537.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342808623.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342823646.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342714590.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342402669.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342418663.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342293728.png)
9分
(3)解:由題
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342886676.png)
,且由(1)知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342901589.png)
.∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342901667.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342917565.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342933527.png)
. 因
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342121395.png)
是底面圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342137292.png)
的直徑,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342979529.png)
,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342995531.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342449680.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342465396.png)
為四棱錐的高.設圓柱高為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021343042313.png)
,底半徑為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021343058260.png)
,
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021343073734.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240213430891298.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021343104468.png)
:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021343120474.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021342340485.png)
. 14分
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823234826293267.png)
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;
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823234826464414.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823234826480391.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823234826495440.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823234826527422.png)
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823234826542403.png)
;
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823234826573364.png)
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;
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