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【題目】某城市響應城市綠化的號召, 計劃建一個如圖所示的三角形 形狀的主題公園,其中一邊利用現成的圍墻, 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,如何設計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)邊長均為米時,所用材料長度最短為

【解析】

試題分析:(1)在中,由余弦定理,得,再由正弦定理得,

,進而可求解求滿足的關系式;(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,當且僅當時,等號成立.

所以,所以,故當邊長均為 米時,所用材料長度最短為 米.

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