【題目】某人上午時(shí),乘摩托艇以勻速從港出發(fā)到距的港去,然后乘汽車以勻速自港向距的市駛?cè)ィ畱?yīng)該在同一天下午至點(diǎn)到達(dá)市.設(shè)乘坐汽車、摩托艇去目的地所需要的時(shí)間分別是.
(1)作圖表示滿足上述條件的范圍;
(2)如果已知所需的經(jīng)費(fèi)(元),那么分別是多少時(shí)最? 此時(shí)需花費(fèi)多少元?
【答案】(1);(2)的最小值為元.
【解析】試題分析:(1)依題意得解出所滿足的范圍,作出符合條件的圖形;(2)先確定目標(biāo)函數(shù),由題意得得出,由(1)中圖可知,當(dāng)先經(jīng)過點(diǎn)時(shí),目標(biāo)函數(shù)取得最小值,即可求解結(jié)論.
試題解析:(1)依題意得
①
由于乘汽車、摩托艇所需的時(shí)間和應(yīng)在至個(gè)小時(shí)之間,即②
因此,滿足①②的點(diǎn)的存在范圍是圖中陰影部分(包括邊界)
(2),上式表示斜率為的直線,
當(dāng)動直線通過圖中的陰影部分區(qū)域(包括邊界),通過點(diǎn)時(shí), 值最。
由得,即當(dāng)時(shí), 最小.
此時(shí), 的最小值為元.
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