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<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">如图所示，小明家小区空地上有两棵笔直的树<i>CD</i>、<i>EF</i>．一天，他在<i>A</i>处测得树顶<i>D</i>的仰角∠<i>DAC</i>＝30°，在<i>B</i>处测得树顶<i>F</i>的仰角∠<i>FBE</i>＝45°，线段<i>BF</i>恰好经过树顶<i>D</i>．已知<i>A</i>、<i>B</i>两处的距离为2米，两棵树之间的距离<i>CE</i>＝3米，<i>A</i>、<i>B</i>、<i>C</i>、<i>E</i>四点在一条直线上，求树<i>EF</i>的高度．（</span></span></span></span></span><img height="18" src="/uploads/meth/20210402/a49bac72800d17903316394b06aa1302.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">≈1.7，</span></span></span></span></span><img height="18" src="/uploads/meth/20210402/97d47c060a9631f6a2dd103d5e4a026f.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">≈1.4，结果保留一位小数）</span></span></span></span></span>                                                                                                                                               <img src="/uploads/meth/20210402/2b4e5b2e157158cb17cb49da457ab816.png" style="float: right; width: 162px; height: 144px;" />

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<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">解：设<i>CD</i>＝<i>xm</i>，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">在Rt△<i>BCD</i>中，∵∠<i>DBC</i>＝45°，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴<i>BC</i>＝<i>CD</i>＝<i>x</i>，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">在Rt△<i>DAC</i>中，∵∠<i>DAC</i>＝30°，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴tan∠<i>DAC</i>＝</span></span></span></span></span><img height="35" src="/uploads/meth/20210402/c0dac29a52de679b222620aa8e72b903.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴<i>x</i>+2＝</span></span></span></span></span><img height="18" src="/uploads/meth/20210402/349d63e8257b9d40c941e5646e10f4bd.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><i><span style="line-height:150%"><span style="font-family:宋体">x</span></span></i><span style="line-height:150%"><span style="font-family:宋体">，解得<i>x</i>＝</span></span></span></span></span><img height="18" src="/uploads/meth/20210402/9e3a26a7f188f561d9451c811402f64d.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">+1，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴<i>BC</i>＝<i>CD</i>＝</span></span></span></span></span><img height="18" src="/uploads/meth/20210402/dc521bf4a38270c6098a2cd301f2fd3a.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">+1，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">在Rt△<i>FBE</i>中，∵∠<i>DBC</i>＝45°，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴<i>FE</i>＝<i>BE</i>＝<i>BC</i>+<i>CE</i>＝</span></span></span></span></span><img height="18" src="/uploads/meth/20210402/733fb30640abf6d78e88bfcca73f1fa1.png" width="20" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">+1+3≈5.7（<i>m</i>）．</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="font-family:宋体">答：树<i>EF</i>的高度约为5.7<i>m</i>．</span></span>

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