题目内容
如图,圆柱
内有一个三棱柱
,三棱柱的底面为圆柱底面的内接三角形,且AB是圆O直径。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231438556034904.jpg)
(Ⅰ)证明:平面
平面
;
(Ⅱ)设AB=
,在圆柱
内随机选取一点,记该点取自于三棱柱
内的概率为
。
(i)当点C在圆周上运动时,求
的最大值;
(ii)记平面
与平面
所成的角为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855775200.gif)
,当
取最大值时,求
的值。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855572259.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855588435.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231438556034904.jpg)
(Ⅰ)证明:平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855619355.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650337.gif)
(Ⅱ)设AB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855572259.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855588435.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855713202.gif)
(i)当点C在圆周上运动时,求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855713202.gif)
(ii)记平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855744333.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855759299.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855775200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855791411.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855713202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855837270.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855837233.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231438558691567.gif)
(Ⅰ)因为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650264.gif)
平面ABC,
平面ABC,所以![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855900108.gif)
,
因为AB是圆O直径,所以![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855947245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855900108.gif)
,又![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856009263.gif)
,所以![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855947245.gif)
平面
,
而
平面
,所以平面
平面
。
(Ⅱ)(i)设圆柱的底面半径为
,则AB=
,故三棱柱
的体积为
=
,又因为
,
所以
=
,当且仅当
时等号成立,
从而
,而圆柱的体积
,
故
=
当且仅当
,即
时等号成立,
所以
的最大值是
。
(ii)由(i)可知,
取最大值时,
,于是以O为坐标原点,建立空间直角坐标系
(如图),则C(r,0,0),B(0,r,0),
(0,r,2r),
因为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855947245.gif)
平面
,所以
是平面
的一个法向量,
设平面
的法向量
,由
,故
,
取
得平面
的一个法向量为
,因为
,
所以
。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855900108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855915268.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855900108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855947245.gif)
因为AB是圆O直径,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855947245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855900108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855993242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856009263.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856025312.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855947245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855900108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855744333.gif)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855915268.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650337.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855619355.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855650337.gif)
(Ⅱ)(i)设圆柱的底面半径为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856165132.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856181322.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855588435.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856212633.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856243333.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856259616.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856259716.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856290218.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856305532.gif)
从而
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856321405.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856337517.gif)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855713202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856368639.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856305532.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856383340.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855713202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855837233.gif)
(ii)由(i)可知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855713202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856383340.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856477390.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856493226.gif)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855947245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855900108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855744333.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856571400.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855744333.gif)
设平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855759299.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856633455.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231438566491081.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856664530.gif)
取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856680215.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143855759299.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856695375.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823143856727352.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231438567421570.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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