【題目】某城市響應(yīng)城市綠化的號召， 計(jì)劃建一個(gè)如圖所示的三角形 形狀的主題公園,其中一邊利用現(xiàn)成的圍墻， 長度為米， 另外兩邊使用某種新型材料圍成， 已知單位均為米）．

（1）求 滿足的關(guān)系式（指出的取值范圍）；

（2）在保證圍成的是三角形公園的情況下，如何設(shè)計(jì)能使所用的新型材料總長度最短？ 最短長度是多少？

【答案】（1），，；（2）當(dāng)邊長均為米時(shí)，所用材料長度最短為米．

【解析】

試題分析：（1）在中,由余弦定理，得，再由正弦定理得，

，進(jìn)而可求解求滿足的關(guān)系式；（2）要使所用的新型材料總長度最短只需最小，由（1）知，，利用基本不等式，即可求解結(jié)論.

試題解析：（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總長度最短只需最小，由（1）知，，由于，當(dāng)且僅當(dāng)時(shí)，等號成立．

所以，所以，故當(dāng)邊長均為 米時(shí)，所用材料長度最短為 米．

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