數學題目 - 推理遊戲

※ 引述《EIORU ()》之銘言：
: 一個25m深水井,
: | |
: d| |
: | |a
: | e |
: | |
: b一一一c
: 斜放了兩個長度為20m(cd相連線段)和15m(ab相連線段)的梯子
: 若此兩個梯子在e的地方交叉
: 已知e點距離井口17m
: 求此井的寬度(bc線段長), 以及角aed的大小
由題意知e點離井底25-17=8m

由e向井底作垂直 垂足為h

則 de:ec = bh:hc = be:ea (由db//eh//ac知)

設井寬x 上述相等的比de/ec = bh/hc = be/ea = r

則 db:eh = bc:hc = (bh+hc):hc = (r+1):1
eh:ac = bh:bc = bh:(bh+hc) = 1:(1+1/r) = r:(r+1)

故eh = db/(r+1) = [r/(r+1)]*ac

又 db^2 = dc^2 - bc^2 = 20^2 - x^2
ac^2 = ab^2 - bc^2 = 15^2 - x^2

所以 20^2 - x^2 = [eh(r+1)]^2
15^2 - x^2 = [eh(r+1)/r]^2

20^2 - [eh(r+1)]^2 = 15^2 - [eh(r+1)/r]^2

代入eh=8

20^2 - [8(r+1)]^2 = 15^2 - [8(r+1)/r]^2

400 - 64(r^2+2r+1) = 225 - 64(1+2/r+1/r^2)

整理可得r的四次式

由此解出r 代回上式可解出x 即bc長

再由三角形ebc可求出角bec 即角aed之大小

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詳細數字等我回去開軟體來算= = 手解四次方程很累人的...
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回來開了軟體來算

上面的四次式 有兩實根兩虛根 實根一正一負

顯然這裡的r只能是正的 所以僅有唯一合理解 約為1.38675

代回得x約為5.95132 即為bc長

然後eb為ab的r/(r+1) = 15 * r/(r+1) = 8.71531

ec為cd的1/(r+1) = 20 * 1/(r+1) = 8.37958

由餘弦定理得 cos 角bec = (eb^2 + ec^2 - bc^2)/2*eb*ec = 0.758283

得該角約為 0.71 弳 ≒ 40.68 度
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'Oh, Harry, dont't you see?' Hermione breathed. 'If she could have done
one thing to make absolutely sure that every single person in this school
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---'Harry Potter and the order of the phoenix', P513

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