题目内容
(A题)小明家准备建造长为28米的蔬菜大棚,示意图如图1.它的横截面为如图2所示的四边形ABCD,已知AB=3米,BC=6米,∠BCD=45°,AB⊥BC,D到BC的距离DE为1米.矩形棚顶ADD′A′及矩形DCC′D′由钢架及塑料薄膜制作,造价为每平方米120元,其它部分(保温墙体等)造价共9250元,则这个大棚的总造价为多少元?(精确到1元)(下列数据可供参考
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_ST/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_ST/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_ST/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_ST/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_ST/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_ST/images5.png)
(B题)如图,河边有一条笔直的公路l,公路两侧是平坦的草地.在数学活动课上,老师要求测量河对岸B点到公路的距离,请你设计一个测量方案.要求:
(1)列出你测量所使用的测量工具;
(2)画出测量的示意图,写出测量的步骤;
(3)用字母表示测得的数据,求出B点到公路的距离.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_ST/images6.png)
【答案】分析:(A)如图所示,过D作DF⊥AB于F,墙体费用已知为9250元,因此必须求出薄膜费用,而面积是关键,由DE=BF=1,DF=BE,∠BCD=45°,可得CE=1,利用勾股定理知CD=
,又BC=6,那么DF=BE=5.在Rt△AFD中,AF=2,DF=5,∴AD=
=5.39,∴塑料薄膜总面积为(
)×28,由此可以求出总造价了;
(B)(1)测角器,尺子
(2)得到任意一锐角三角形,有两角在公路上,量得在公路上的两锐角的大小及线段长,构造直角三角形,利用相应三角函数求解即可;
(3)利用公共边及相应的三角函数分别求得AC,BC长,让它们相加等于CD长求解即可.
解答:
解:(A)如图,过D作DF⊥AB于F.
∵AB⊥BC,
∴DF∥BC,
又∵DE⊥BC,
∴DE∥AB,
∴四边形BEDF为矩形,
∴DE=BF=1,DF=BE,
又∵∠BCD=45°,
∴CE=1,CD=
,
又BC=6,
∴DF=BE=5,
在Rt△AFD中,AF=2,DF=5,
∴AD=
=
=5.39,
∴S四边形ADD'A'=
×28≈150.9,
S四边形DCC'D'=
×28≈39.5,
∴总造价为(150.9+39.5)×120+9250≈32098(元).
(B)(1)测角器、尺子;
(2)测量示意图见图;
测量步骤:
①在公路上取两点C,D,使∠BCD,∠BDC为锐角;
②用测角器测出∠BCD=α,∠BDC=β;
③用尺子测得CD的长,记为m米;
④计算求值.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/images9.png)
(3)解:设B到CD的距离为x米,
作BA⊥CD于点A,在△CAB中,x=CAtanα,
在△DAB中,x=ADtanβ,
∴CA=
,AD=
,
∵CA+AD=m,
∴
,
∴
.
∴B点到公路的距离是
.
点评:解此题的关键是把实际问题转化为数学问题,把实际问题抽象到解直角三角形中,利用三角函数解答.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/2.png)
(B)(1)测角器,尺子
(2)得到任意一锐角三角形,有两角在公路上,量得在公路上的两锐角的大小及线段长,构造直角三角形,利用相应三角函数求解即可;
(3)利用公共边及相应的三角函数分别求得AC,BC长,让它们相加等于CD长求解即可.
解答:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/images3.png)
∵AB⊥BC,
∴DF∥BC,
又∵DE⊥BC,
∴DE∥AB,
∴四边形BEDF为矩形,
∴DE=BF=1,DF=BE,
又∵∠BCD=45°,
∴CE=1,CD=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/3.png)
又BC=6,
∴DF=BE=5,
在Rt△AFD中,AF=2,DF=5,
∴AD=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/5.png)
∴S四边形ADD'A'=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/6.png)
S四边形DCC'D'=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/7.png)
∴总造价为(150.9+39.5)×120+9250≈32098(元).
(B)(1)测角器、尺子;
(2)测量示意图见图;
测量步骤:
①在公路上取两点C,D,使∠BCD,∠BDC为锐角;
②用测角器测出∠BCD=α,∠BDC=β;
③用尺子测得CD的长,记为m米;
④计算求值.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/images9.png)
(3)解:设B到CD的距离为x米,
作BA⊥CD于点A,在△CAB中,x=CAtanα,
在△DAB中,x=ADtanβ,
∴CA=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/8.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/9.png)
∵CA+AD=m,
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/10.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/11.png)
∴B点到公路的距离是
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103102104895177486/SYS201311031021048951774025_DA/12.png)
点评:解此题的关键是把实际问题转化为数学问题,把实际问题抽象到解直角三角形中,利用三角函数解答.
![](http://thumb2018.1010pic.com/images/loading.gif)
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