题目内容

(本题满分7分)将一条长为20cm的铁丝剪成两段,并以每一段铁丝的长度为周长做成一个正方形.

(1)要使这两个正方形的面积之和等于,那么这段铁丝剪成两段后的长度分别是多少?

(2)两个正方形的面积之和可能等于吗?若能,求出两段铁丝的长度;若不能,请说明理由.

 

解:设这段铁丝被分成两段后,围成正方形.其中一个正方形的边长为,则另一个正方形的边长为.·········································································································· 1分

 

依题意列方程得:

,················································································· 3分

解方程得:

因此这段铁丝剪成两段后的长度分别是4cm、16cm.·········································· 5分

(2)两个正方形的面积之和不可能等于.理由:

设两个正方形的面积和为,则:

,····························································· 7分

 

,当的最小值

 

两个正方形的面积之和不可能等于.·················································· 9分

(另解:由(1)可知:,···················································· 6分

化简后得:

方程无实数解,

所以两个正方形的面积之和不可能等于.)·············································· 9分

解析:略

 

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