题目

A display case is in the shape of a rectangular box with a square base.Suppose the volume is 21 cubic ft and it costs $1 per square ft.to build the glass top and $0.50 per sq.ft.to build the sides and base.If x is the length of one side of the base,what value should x have to minimize the cost?Round your answer to two decimal places.

提问时间：2020-10-27

答案

题目看得懂么?
换成数学语言就是一个长方体,底是正方形,面积为21,顶部费用为每单位1,侧面和底部费用为每单位0.5,现设底部长度为x,求x,使费用最低（结果保留两位小数）.
底部正方形变长为x,设长方体高为h,则x²h=21,即h=21/x²
费用C=0.5*（x²+4xh）+1*x²=1.5x²+2xh=1.5x²+42/x
求导,得C‘=3x-42/x² C’=0,得x=14的三次方根
x （0,14的三次方根） 14的三次方根 14的三次方根
C‘ - 0 +
C ↓ ↑
即x=14的三次方根时,C取得最小值
即x=2.41时,费用最低,大约为23.24