[题目]某城市响应城市绿化的号召. 计划建一个如图所示的三角形形状的主题公园,其中一边利用现成的围墙. 长度为米. 另外两边使用某种新型材料围成. 已知单位均为米)．(1)求满足的关系式(指出的取值范围), (2)在保证围成的是三角形公园的情况下.如何设计能使所用的新型材料总长度最短? 最短长度是多少? 题目和参考答案——青夏教育精英家教网—

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【题目】某城市响应城市绿化的号召，计划建一个如图所示的三角形形状的主题公园,其中一边利用现成的围墙，长度为米，另外两边使用某种新型材料围成，已知单位均为米）．

（1）求满足的关系式（指出的取值范围）；

（2）在保证围成的是三角形公园的情况下，如何设计能使所用的新型材料总长度最短？最短长度是多少？

【答案】（1），，；（2）当边长均为米时，所用材料长度最短为米．

【解析】

试题分析：（1）在中,由余弦定理，得，再由正弦定理得，

，进而可求解求满足的关系式；（2）要使所用的新型材料总长度最短只需最小，由（1）知，，利用基本不等式，即可求解结论.

试题解析：（1）在中,由余弦定理，得，即，

由正弦定理，得，

同理.

（2）要使所用的新型材料总长度最短只需最小，由（1）知，，由于，当且仅当时，等号成立．

所以，所以，故当边长均为米时，所用材料长度最短为米．

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