2011·吉林·一模
1 . 如图所示,现有一张边长为
的正三角形纸片
,在三角形的三个角沿图中虚线剪去三个全等的四边形
,
,
(剪去的四边形均有一组对角为直角),然后把三个矩形
,
,
折起,构成一个以
为底面的无盖正三棱柱.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/16/77d9aa85-32ff-4c15-9fae-3525d8650737.png?resizew=165)
(1)若所折成的正三棱柱的底面边长与高之比为3,求该三棱柱的高;
(2)求所折成的正三棱柱的体积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6293f10977e09f6bf2be1c49fb2c874.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b6e12f10fb9547f91fa055667d44cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698652bebc18c54983503d0a5ddb0bb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/813c279752688a5b0973b93812e35d9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a16460e02999d0f8c2174df3c0cd21fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25a6c0a9cd5dfdca74cdcf9186bde58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b9c45578c43f597584d1b38730917e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b16cff607cdc2d69afc70dc778acbb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/16/77d9aa85-32ff-4c15-9fae-3525d8650737.png?resizew=165)
(1)若所折成的正三棱柱的底面边长与高之比为3,求该三棱柱的高;
(2)求所折成的正三棱柱的体积的最大值.
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