1 . 某小区有块绿地,绿地的平面图大致如下图所示,并铺设了部分人行通道.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/14/c506cd7e-0c97-49b3-acfb-a25b782ba60b.png?resizew=168)
为了简单起见,现作如下假设:
假设1:绿地是由线段
,
,
,
和弧
围成的,其中
是以
点为圆心,圆心角为
的扇形的弧,见图1;
假设2:线段
,
,
,
所在的路行人是可通行的,圆弧
暂时未修路;
假设3:路的宽度在这里暂时不考虑;
假设4:路用线段或圆弧表示,休息亭用点表示.
图1-图3中的相关边、角满足以下条件:
直线
与
的交点是
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68fdb2b9d6a4a54ed1328c5b3adcf7b6.png)
,
.
米.
小区物业根据居民需求,决定在绿地修建一个休息亭.根据不同的设计方案解决相应问题,结果精确到米.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/15/5e27693b-d213-4a76-b176-210f478b2a83.png?resizew=565)
(1)假设休息亭建在弧
的中点,记为
,沿
和线段
修路,如图2所示.求
的长;
(2)假设休息亭建在弧
上的某个位置,记为
,作
交
于
,作
交
于
.沿
、线段
和线段
修路,如图3所示.求修建的总路长
的最小值;
(3)请你对(1)和(2)涉及到的两种设计方案做个简明扼要的评价.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/14/c506cd7e-0c97-49b3-acfb-a25b782ba60b.png?resizew=168)
为了简单起见,现作如下假设:
假设1:绿地是由线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0450d195423158b2963b517f0425ba57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0450d195423158b2963b517f0425ba57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0211da37e92f915e781691296578ba0.png)
假设2:线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0450d195423158b2963b517f0425ba57.png)
假设3:路的宽度在这里暂时不考虑;
假设4:路用线段或圆弧表示,休息亭用点表示.
图1-图3中的相关边、角满足以下条件:
直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68fdb2b9d6a4a54ed1328c5b3adcf7b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72c4340dcffb0783d118a587e5352a2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79f2eff72d0b77615ee85ec9676a8d16.png)
小区物业根据居民需求,决定在绿地修建一个休息亭.根据不同的设计方案解决相应问题,结果精确到米.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/15/5e27693b-d213-4a76-b176-210f478b2a83.png?resizew=565)
(1)假设休息亭建在弧
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0450d195423158b2963b517f0425ba57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0450d195423158b2963b517f0425ba57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/350e954f629c1901a5cec03558319e46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/350e954f629c1901a5cec03558319e46.png)
(2)假设休息亭建在弧
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0450d195423158b2963b517f0425ba57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a470095e295c734a2f368cc6baf1b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acd378d3805a3d8295aec88bf3684f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a4fdec4ee1fc83fdd6b4c324755d95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a66f7735b5a72e63461fbf457e7e3977.png)
(3)请你对(1)和(2)涉及到的两种设计方案做个简明扼要的评价.
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2023-04-13更新
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4卷引用:上海市闵行区上海外国语大学闵行外国语中学2024届高三上学期期中数学试题