Question

（本題滿分7分）將一條長為20cm的鐵絲剪成兩段，并以每一段鐵絲的長度為周長做成一個正方形．
（1）要使這兩個正方形的面積之和等于，那么這段鐵絲剪成兩段后的長度分別是多少？
（2）兩個正方形的面積之和可能等于嗎？若能，求出兩段鐵絲的長度；若不能，請說明理由．

Answer 1

解：設這段鐵絲被分成兩段后，圍成正方形．其中一個正方形的邊長為，則另一個正方形的邊長為．············································································· 1分
依題意列方程得：
，················································································· 3分
解方程得：，
因此這段鐵絲剪成兩段后的長度分別是4cm、16cm．·········································· 5分
（2）兩個正方形的面積之和不可能等于．理由：
設兩個正方形的面積和為，則：
，····························································· 7分
，當，的最小值．
兩個正方形的面積之和不可能等于．·················································· 9分
（另解：由（1）可知：，···················································· 6分
化簡后得：，
，
方程無實數解，
所以兩個正方形的面積之和不可能等于．）·············································· 9分

解析