作业帮哥哥姐姐 A farmer wants to fence an area of 1.5 million square f
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哥哥姐姐
A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle.How can he do this so as to minimize the cost of the fence?(Give the dimensions in increasing order.)
A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle.How can he do this so as to minimize the cost of the fence?(Give the dimensions in increasing order.)
比如你的field的长是x,宽是y,则你需要的总长度是2x+3y,或者是3x+2y,比如你的cost是a,则你需要考虑的是让(2x+3y)a最小或者(3x+2y)a最小,其实就是让2x+3y最小或者让3x+2y最小
根据均值不等式,3x+2y≥2√6xy=2√9×10^6=6×10^3,所以当3x=2y时,花费最小
同理,如果是2x+3y,最小值也是6×10^3,即2x=3y的时候,花费最小
综上：如果你的field的长宽比例是3：2,(即长1500feet,宽1000feet),且中间那块儿fence的长度跟宽一样长,即1000feet,那么花费最小,或者是,如果你的field长宽比例是2：3,(即长1000feet,宽1500feet),且中间那块儿fence的长度跟长一样长,即1000feet,那么花费最小
根据均值不等式,3x+2y≥2√6xy=2√9×10^6=6×10^3,所以当3x=2y时,花费最小
同理,如果是2x+3y,最小值也是6×10^3,即2x=3y的时候,花费最小
综上：如果你的field的长宽比例是3：2,(即长1500feet,宽1000feet),且中间那块儿fence的长度跟宽一样长,即1000feet,那么花费最小,或者是,如果你的field长宽比例是2：3,(即长1000feet,宽1500feet),且中间那块儿fence的长度跟长一样长,即1000feet,那么花费最小
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