题目内容
如图,一单杆高2.2m,两立柱之间的距离为1.6m,将一根绳子的两端栓于立柱与铁杠结合处,绳子自然下垂呈抛物线状.(1)一身高0.7m的小孩站在离立柱0.4m处,其头部刚好触上绳子,求绳子最低点到地面的距离;
(2)为供孩子们打秋千,把绳子剪断后,中间系上一块长为0.4米的木板,除掉系木板用去的绳子后,两边的绳子正好各为2米,木板与地面平行,求这时木板到地面的距离.(供选用数据:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_ST/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_ST/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_ST/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_ST/images3.png)
【答案】分析:(1)设二次函数为y=ax2+c,利用待定系数法求出a,c的值然后可求出绳子最低点到地面的距离.
(2)本题要靠辅助线的帮助求出AG的值.然后根据勾股定理求出EG的值.
解答:解:(1)如图,建立直角坐标系,设二次函数为:y=ax2+c
∵D(-0.4,0.7),B(0.8,2.2)
∴![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_DA/0.png)
∴![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_DA/1.png)
∴绳子最低点到地面的距离为0.2米.
(2)分别作EG⊥AB于G,E、FH⊥AB于H,
AG=
(AB-EF)=
(1.6-0.4)=0.6
在Rt△AGE中,AE=2,EG=
≈1.9
∴2.2-1.9=0.3(米)
∴木板到地面的距离约为0.3米.
点评:本题综合考查了二次函数与图象结合实际应用的有关知识,难度中上.
(2)本题要靠辅助线的帮助求出AG的值.然后根据勾股定理求出EG的值.
解答:解:(1)如图,建立直角坐标系,设二次函数为:y=ax2+c
∵D(-0.4,0.7),B(0.8,2.2)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_DA/0.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_DA/1.png)
∴绳子最低点到地面的距离为0.2米.
(2)分别作EG⊥AB于G,E、FH⊥AB于H,
AG=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_DA/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_DA/3.png)
在Rt△AGE中,AE=2,EG=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103101618660443662/SYS201311031016186604436024_DA/4.png)
∴2.2-1.9=0.3(米)
∴木板到地面的距离约为0.3米.
点评:本题综合考查了二次函数与图象结合实际应用的有关知识,难度中上.
![](http://thumb2018.1010pic.com/images/loading.gif)
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