题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-13239.png)
(1)若最大拱高h为6 m,则隧道设计的拱宽
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-62534.png)
(2)若要使隧道上方半椭圆部分的土方工程 量最小,则应如何设计拱高h和拱宽
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-62534.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-84784.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-82513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-39043.png)
(3)为了使隧道内部美观,要求在拱线上找两个点M、N,使它们所在位置的高度恰好是限高5m,现以M、N以及椭圆的左、右顶点为支点,用合金钢板把隧道拱线部分连接封闭,形成一个梯形,若l=30m,梯形两腰所在侧面单位面积的钢板造价是梯形顶部单位面积钢板造价的
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-66965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-28168.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-76473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-95925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-66965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-90462.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-38492.png)
解析
试题分析:(1)先建立直角坐标系,找到对应椭圆方程再把b=h-3=3与点P坐标代入椭圆方程,即可求出隧道设计的拱宽l是多少;
(2)转化为求半椭圆的面积最小值问题,对椭圆方程用基本不等式即可求出对应的半椭圆面积以及满足要求的拱高h和拱宽l.
(3)先求出总造价的表达式,再利用导函数研究其最值即可.
试题解析:解:(1)如下图建立直角坐标系,则点P(10,2),椭圆方程为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-84784.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-82513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-34040.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-76473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-76473.png)
(2)要使隧道上方半椭圆部分的土方工程量最小,由柱体的体积公式可知:只需半椭
圆的面积最小即可。
由椭圆方程
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-84784.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223700-82513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-14470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223703-17594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-14470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223703-17594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223703-16877.png)
所以半椭圆面积S=
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223703-20123.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223703-86361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-14470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223703-17594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223704-76683.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-66965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-95925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-66965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-95925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-95925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223701-66965.png)
(3)设
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223704-73444.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223704-90660.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223704-13107.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-95925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223705-28258.png)
=2
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223705-40167.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223705-85779.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223705-17028.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223705-77213.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223706-93980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223706-93980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223704-90660.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-90462.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223706-14812.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223702-38492.png)
核心考点
试题【(13分)如图,某隧道设计为双向四车道,车道总宽20m,要求通行车辆限高5m,隧道全长2.5km,隧道的两侧是与地面垂直的墙,高度为3米,隧道上部拱线近似地看成】;主要考察你对曲线与方程的应用等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223649-64205.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223649-27224.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223649-53585.jpg)
(1)求椭圆C的方程;
(2)已知P(2,3)、Q(2,-3)是椭圆上的两点,A,B是椭圆上位于直线PQ两侧的动点,
①若直线AB的斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223649-64205.png)
②当A、B运动时,满足
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223649-71083.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223649-44864.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-93117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-48728.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-98666.png)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-93117.png)
(2)若以
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-94203.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-66341.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-93117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-72260.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-82057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223638-72240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223639-80335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223623-92539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223623-83469.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223623-73827.png)
(1)求椭圆C的方程:
(2)若P是椭圆上异于A,B的动点,连结AP,PB并延长,分别与右准线
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223623-72332.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-37210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-48140.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-90128.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-59357.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-37210.png)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-37210.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-60959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-37210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-60959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-78744.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223608-61942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223607-37210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223608-94522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223608-19832.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223608-29606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-56778.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-36639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-24588.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-99155.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-24588.png)
(1)求椭圆
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-24588.png)
(2)过
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-51714.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223553-27083.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223552-24588.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223553-54720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191025/20191025223553-91644.png)
若存在其最大值及此时的直线方程;若不存在,请说明理由.
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