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解题方法
1 . 某数学建模小组研究挡雨棚(图1),将它抽象为柱体(图2),底面
与
全等且所在平面平行,
与
各边表示挡雨棚支架,支架
、
、
垂直于平面
.雨滴下落方向与外墙(所在平面)所成角为
(即
),挡雨棚有效遮挡的区域为矩形
(
、
分别在
、
延长线上).
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/8b926056-0b86-4ce8-9ffa-f4024886010a.png?resizew=361)
(1)挡雨板(曲面
)的面积可以视为曲线段
与线段
长的乘积.已知
米,
米,
米,小组成员对曲线段
有两种假设,分别为:①其为直线段且
;②其为以
为圆心的圆弧.请分别计算这两种假设下挡雨板的面积(精确到0.1平方米);
(2)小组拟自制
部分的支架用于测试(图3),其中
米,
,
,其中
,求有效遮挡区域高
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b16cff607cdc2d69afc70dc778acbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56150248dd4b787a2013311e4737e93f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2558d8d867325a0460ec7f638d5dfd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abaeba15f3abdd877bc701af52c5cd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cabc3303519ac16fc998913ad9f349c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/8b926056-0b86-4ce8-9ffa-f4024886010a.png?resizew=361)
(1)挡雨板(曲面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58cc6184b191e6da43911e701121517e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/196bd4d803ca8836b69fa54c6daf94c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a72cc14d6ce21597bb21db50bf705422.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92535536bd3c2761724fd058427f95a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f2a8d0c77c54fdf9aa97978f8e5dc0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(2)小组拟自制
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6fe0ece7af6e56ad40a25e8e1b8f97a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72c4340dcffb0783d118a587e5352a2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e30e586a8505faad251cc182676e0390.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a2d256a64154ccaae236e5a8a1012eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
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2023-12-13更新
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2 . 某水产养殖户承包一片靠岸水域.如图,
、
为直线岸线,
米,
米,
,该承包水域的水面边界是某圆的一段弧
,过弧
上一点
按线段
和
修建养殖网箱,已知
.
![](https://img.xkw.com/dksih/QBM/2021/12/21/2877362589736960/2878134138093568/STEM/5cf2c44e260345edaf70618b57930d9a.png?resizew=178)
(1)求岸线上点
与点
之间的直线距离;
(2)如果线段
上的网箱每米可获得40元的经济收益,线段
上的网箱每米可获得30元的经济收益.记
,则这两段网箱获得的经济总收益最高为多少?(精确到元)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b711d09cf350b68728ace265fc88d7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/602773cbc4969675b9e28c8fa8f77d3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d162c29b1e484cfc87350dd68f00b85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49c27ed2753bc9782bc9fa26755eb582.png)
![](https://img.xkw.com/dksih/QBM/2021/12/21/2877362589736960/2878134138093568/STEM/5cf2c44e260345edaf70618b57930d9a.png?resizew=178)
(1)求岸线上点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)如果线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/513cb0e220a0fed33454151e303bcbe7.png)
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