【題目】要建造一個(gè)容積為1 600立方米，深為4米的長(zhǎng)方體無蓋蓄水池，池壁的造價(jià)為每平方米200元，池底的造價(jià)為每平方米100元．

（1）把總造價(jià)y元表示為池底的一邊長(zhǎng)x米的函數(shù)；

（2）由于場(chǎng)地原因，蓄水池的一邊長(zhǎng)不能超過20米，問蓄水池的這個(gè)底邊長(zhǎng)為多少時(shí)總造價(jià)最低？總造價(jià)最低是多少？

【答案】（1）y＝1 600（x＋）＋40 000，x∈（0，＋∞）；（2）20，104 000

【解析】（1）由已知得池底的面積為＝400（平方米），底面的另一邊長(zhǎng)為米，則池壁的面積為2×4×（x＋）平方米．

所以y＝1 600（x＋）＋40 000，x∈（0，＋∞）．

（2）由（1）知y＝1 600（x＋）＋40 000（0＜x≤20），

設(shè)0＜x1＜x2≤20，則

y1－y2＝1 600（x1＋）－1 600（x2＋）

＝1 600[（x1－x2）＋]

＝1 600（x1－x2）（1－）．

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

從而這個(gè)函數(shù)在（0，20]上是減函數(shù)，故當(dāng)x＝20時(shí)，ymin＝10 4000.

所以當(dāng)池底是邊長(zhǎng)為20米的正方形時(shí)，總造價(jià)最低，為104 000元．

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