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【题目】建造一个容积为1 600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】1y=1 600x+40 000,x∈0,+∞220,104 000

【解析】1由已知池底的面积400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=1 600x+40 000,x∈0,+∞

21知y=1 600x+40 0000x≤20

设0x1x220,则

y1-y2=1 600x1-1 600x2

=1 600[x1-x2]

=1 600x1-x2)(1-

∵0x1x220,∴x1-x201-0,得y1-y20,即y1y2.

从而这个函数在0,20]上是减函数,故当x=20时,ymin10 4000.

所以当池底是边长为20米的正方形时,总造价最低104 000元.

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