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【题目】某人上午时, 乘摩托艇以匀速港出发到距港去, 然后乘汽车以匀速港向距市驶去.应该在同一天下午点到达市. 设乘坐汽车、 摩托艇去目的地所需要的时间分别是.

(1)作图表示满足上述条件的范围

(2)如果已知所需的经费(元),那么分别是多少时最小? 此时需花费多少元?

【答案】(1);(2)的最小值为

【解析】

试题分析:(1)依题意得 解出所满足的范围,作出符合条件的图形;(2)先确定目标函数,由题意得得出,由(1)中图可知,当先经过点时,目标函数取得最小值,即可求解结论.

试题解析:(1)依题意得

由于乘汽车、摩托艇所需的时间和应在个小时之间,

因此,满足①②的点的存在范围是图中阴影部分(包括边界)

(2)上式表示斜率为的直线,

当动直线通过图中的阴影部分区域(包括边界),通过点时,最小.

,即当时,最小.

此时,的最小值为 元.

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