如图1，已知长方体的长为BC=2cm，宽AC=1cm，高AA′=4cm．一只蚂蚁如果沿长方体的表面从A点爬到B′点，那么 已知长方体的长AC为2cm，宽BC为1cm，高AA'为4cm
\uff082008?\u5929\u6cb3\u533a\u4e8c\u6a21\uff09\u5982\u56fe1\uff0c\u5df2\u77e5\u957f\u65b9\u4f53\u7684\u957f\u4e3aBC=2cm\uff0c\u5bbdAC=1cm\uff0c\u9ad8AA\u2032=4cm\uff0e\uff081\uff09\u4e00\u53ea\u8682\u8681\u5982\u679c\u6cbf\u957f\u65b9\u4f53\u89e3\u7b54\uff1a\u89e3\uff1a\uff081\uff09\u6839\u636e\u9898\u610f\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6700\u77ed\u8def\u5f84\u6709\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5\uff1a\u2460\u6cbfAA\u2032\uff0cA\u2032C\u2032\uff0cC\u2032B\u2032\uff0cB\u2032B\u526a\u5f00\uff0c\u5f97\u56fe\uff081\uff09AB\u20322=AB2+BB\u20322=\uff082+1\uff092+42=25\uff0c\u2461\u6cbfAC\uff0cCC\u2032\uff0cC\u2032B\u2032\uff0cB\u2032D\u2032\uff0cD'A'\uff0cA\u2032A\u526a\u5f00\uff0c\u5f97\u56fe\uff082\uff09AB\u20322=AC2+B\u2032C2=12+\uff084+2\uff092=37\uff0c\u2462\u6cbfAD\uff0cDD\u2032\uff0cB\u2032D\u2032\uff0cC\u2032B\u2032\uff0cC\u2032A\u2032\uff0cAA\u2032\u526a\u5f00\uff0c\u5f97\u56fe\uff083\uff09AB\u20322=AD2+B\u2032D2=22+\uff084+1\uff092=29\uff0c\u7efc\u4e0a\u6240\u8ff0\uff0c\u6700\u77ed\u8def\u5f84\u5e94\u4e3a\uff081\uff09\u6240\u793a\uff0c\u6240\u4ee5AB\u20322=25\uff0c\u5373AB\u2032=5cm\uff0c\u7b54\uff1a\u6700\u77ed\u8def\u5f84\u4e3a\uff081\uff09\u6240\u793a5cm\uff1b\uff082\uff09\u8981\u4fdd\u8bc1\u5e95\u9762\u5706\u6700\u5927\uff0c\u5fc5\u987b\u4f7f\u5f97\u5706\u4e0e\u957f\u65b9\u5f62\u7684\u4e24\u6761\u957f\u8fb9\u76f8\u5207\uff0c\u5219\u6b64\u65f6\u5706\u7684\u534a\u5f84\u957f\u4e3a12\uff0c\u5f53\u5706\u4e0eA\u2032C\u2032\u76f8\u5207\u65f6\uff0c\u2220A\u20320C\u2032\u6700\u5927\uff0c\u6b64\u65f6\u4e3a90\u00b0\uff1b\u5f53\u5706\u4e0eB\u2032D\u2032\u76f8\u5207\u65f6\uff0cE\u4e3a\u5207\u70b9\uff0c\u2220A\u20320C\u2032\u6700\u5c0f\uff0c\u6b64\u65f6\uff0cA\u2032E=12\uff0cOE=32\uff0ctan\u2220A\u20320E=13\u6240\u4ee5\u2220A\u20320E\u224818.4\u00b0\uff0c\u2220A\u20320C\u2032\u224836.8\u00b0\uff0c\u2220A\u2032OC\u2032\u7684\u5ea6\u6570\u8303\u56f4\u4e3a36.8\u00b0\u2264\u2220A\u2032OC\u2032\u226490\u00b0\uff0e
\u4f60\u753b\u4e2a\u957f\u65b9\u4f53\u7684\u5c55\u5f00\u56fe\uff0c\u7136\u540e\u628a\u4e24\u70b9\u76f4\u7ebf\u76f8\u8fde\uff0c\u6839\u636e\u4e09\u89d2\u7b97\u51fa
如果这个蚂蚁很聪明，那么将长方体侧面展开，然后用直线连接A和B'点，可得蚂蚁爬过去的最小路径。根据侧面展开不同，可有三种选择：（1）将面AA‘C’C和面A‘D’B‘C’展开，构成一个6cm*1cm的长方形，这样最短路径为（37的二次方）cm；
（2）将面AA‘D’D和面A‘D’B‘C’展开，构成一个5cm*2cm的长方形，这样最短路径为（29的二次方）cm；
（3）将面AA‘D’D和面BB’D‘D展开，构成一个4cm*3cm的长方形，这样最短路径为（25的二次方）cm=5cm。
（1）展开前面右面由勾股定理得AB2=（2+4）2+12=37；
（2）展开前面上面由勾股定理得AB2=（1+4）2+22=29；
（3）展开左面上面由勾股定理得AB2=（2+1）2+42=25．
侧面展开图
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