【答案】
分析:(1)在三角形AEP中,AP=2,∠A=60°,利用三角函數(shù)可求出AE和PE,即可求出面積;
(2)①此題應(yīng)分情況討論,因?yàn)閮蓚€(gè)動(dòng)點(diǎn)運(yùn)動(dòng)速度不同,所以有點(diǎn)P與點(diǎn)Q都在AB上運(yùn)動(dòng)、點(diǎn)P在BC上運(yùn)動(dòng)點(diǎn)Q仍在AB上運(yùn)動(dòng)、點(diǎn)P和點(diǎn)Q都在BC上運(yùn)動(dòng)三種情況,在每種情況下可利用三角函數(shù)分別求出我們所需要的值,進(jìn)而求解.
②在①的基礎(chǔ)上,首先①求出函數(shù)關(guān)系式之后,根據(jù)t的取值范圍不同函數(shù)最大值也不同.
解答:解:(1)當(dāng)點(diǎn)P運(yùn)動(dòng)2秒時(shí),AP=2cm,由∠A=60°,知AE=1,PE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/0.png)
.(2分)
∴S
△APE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/1.png)
;(4分)
(2)①當(dāng)0≤t<6時(shí),點(diǎn)P與點(diǎn)Q都在AB上運(yùn)動(dòng),如圖所示:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/images2.png)
設(shè)PM與AD交于點(diǎn)G,QN與AD交于點(diǎn)F,
則AQ=t,AF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/2.png)
,QF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/3.png)
t,
AP=t+2,AG=1+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/4.png)
,PG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/5.png)
+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/6.png)
t.
∴此時(shí)兩平行線截平行四邊形ABCD的面積為S=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/7.png)
t+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/8.png)
;(8分)
②當(dāng)6≤t<8時(shí),點(diǎn)P在BC上運(yùn)動(dòng),點(diǎn)Q仍在AB上運(yùn)動(dòng).如圖所示:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/images10.png)
設(shè)PM與DC交于點(diǎn)G,QN與AD交于點(diǎn)F,則AQ=t,AF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/9.png)
,
DF=4-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/10.png)
,QF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/11.png)
t,BP=t-6,CP=10-t,PG=(10-t)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/12.png)
,
而BD=4
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/13.png)
,故此時(shí)兩平行線截平行四邊形ABCD的面積為S=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/14.png)
t
2+10
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/15.png)
t-34
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/16.png)
,(10分)
③當(dāng)8≤t≤10時(shí),點(diǎn)P和點(diǎn)Q都在BC上運(yùn)動(dòng).如圖所示:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/images19.png)
設(shè)PM與DC交于點(diǎn)G,QN與DC交于點(diǎn)F,則CQ=20-2t,QF=(20-2t)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/17.png)
,
CP=10-t,PG=(10-t)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/18.png)
.
∴此時(shí)兩平行線截平行四邊形ABCD的面積為S=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/19.png)
.(14分)
故S關(guān)于t的函數(shù)關(guān)系式為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/20.png)
;
②(附加題)當(dāng)0≤t<6時(shí),S的最大值為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/21.png)
,(1分)
當(dāng)6≤t<8時(shí),S的最大值為6
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/22.png)
,(舍去),(2分)
當(dāng)8≤t≤10時(shí),S的最大值為6
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/23.png)
,(3分)
所以當(dāng)t=8時(shí),S有最大值為6
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103101624222095826/SYS201311031016242220958010_DA/24.png)
.(4分)
(如正確作出函數(shù)圖象并根據(jù)圖象得出最大值,同樣給4分)
點(diǎn)評:此題解答需數(shù)形結(jié)合,把函數(shù)知識和幾何知識緊密聯(lián)系在一起,難易程度適中.