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【题目】要建造一个容积为1600立方米,深为4米的长方体无盖蓄水池,池壁的造价为每平方米200元,池底的造价为每平方米100元.

(1)把总造价y元表示为池底的一边长x米的函数;

(2)由于场地原因,蓄水池的一边长不能超过20米,问蓄水池的这个底边长为多少时总造价最低?总造价最低是多少?

【答案】见解析

【解析】(1)由已知得池底的面积为(平方米),底面的另一边长为米,则池壁的面积为2×4×(x+)平方米.

所以y=1600(x+)+40000,

(2)由(1)知y=1600(x+)+40000(0<x≤20),

设0<x1<x2≤20,则

y1-y21600(x1)-1600(x2

0<x1<x2≤20,x1-x2<0,1-<0,得y1-y2>0,即y1>y2.

从而这个函数在(0,20]上是减函数,故当x=20时,ymin104000.

故当池底是边长为20米的正方形时,总造价最低,为104000元.

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