問題の解答
検索用コード（LaTeX）
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% 例題I3.3.33：2次不等式の文章題（One More）★★
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右の図のような直角二等辺三角形ABCの辺上に頂点をもつ長方形PQRAを作る．長方形の面積が$7\mathrm{~cm}^2$以上$12\mathrm{~cm}^2$以下となるときの$x$の値の範囲を求めよ．
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% 例題I3.3.33の解答（One More）★★
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$\mathrm{BP}=\mathrm{PQ}=x$であるから，
\[
\mathrm{PA}=\mathrm{BA}-\mathrm{BP}=8-x
\]
$x>0,8-x>0$であるから，$0<x<8\cdots(\mathrm{i})$
長方形PQRAの面積は$x(8-x)\mathrm{~cm}^2$が$7\mathrm{~cm}^2$以上$12\mathrm{~cm}^2$であるから，
\[
7\leqq x(8-x)\leqq 12
\]
$7\leqq x(8-x)$より，$x^2-8x+7\leqq 0$
したがって，$(x-1)(x-7)\leqq 0$
ゆえに，$1\leqq x\leqq 7\cdots(\mathrm{ii})$
$x(8-x)\leqq 12$より，$x^2-8x+12\geqq 0$
したがって，$(x-2)(x-6)\geqq 0$
ゆえに，$x\leqq 2,6\leqq x\cdots(\mathrm{iii})$
よって，(i)〜(iii)の共通範囲を求めると，
\[
1\leqq x\leqq 2,6\leqq x\leqq 7
\]
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% 問題I3.3.33：2次不等式の文章題（One More）★★
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長さ40 cmの針金を2つに分け，それぞれを折り曲げて正方形を2つ作る．2つの正方形の面積の和が$52\mathrm{cm}^2$以上になるようにするには，針金をどのように切ればよいか．短い方の針金の長さの範囲を求めよ．
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% 問題I3.3.33の解答（One More）★★
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短い方の針金の長さを$4x\mathrm{cm}$とすると，長い方の針金の長さは，
\[
40-4x=4(10-x)(\mathrm{cm})
\]
$0<4x<20$より，$0<x<5\cdots(\mathrm{i})$
2つの正方形の1辺の長さは，それぞれ$x\mathrm{cm}$と$10-x\mathrm{cm}$であるから，
\[
x^2+(10-x)^2\geqq 52
\]
整理すると，$x^2-10x+24\geqq 0$
したがって，$(x-6)(x-4)\geqq 0$
ゆえに，$x\leqq 4,6\leqq x\cdots(\mathrm{ii})$
(i)，(ii)より，$0<x\leqq 4$
よって，$0<4x\leqq 16$となり，短い方の針金の長さは，$0\mathrm{cm}$より長く，$16\mathrm{cm}$以下であればよい．
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