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