[微積] 最少的材料做成的罐頭

看板Math作者 (雲霧)時間14年前 (2011/07/26 22:56), 編輯推噓2(2推 0噓 9→)
留言11則, 3人參與, 最新討論串1/1
A closest cylindrical can is to hold 500cm^3 of liquid , how should we choose the height and radius to minimize the amount of material needed to manufacture the can? -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 111.249.93.31
推
07/26 22:57, , 1F
目標函數
07/26 22:57, 1F
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07/26 22:57, , 2F
pi*r^2*h = 500
07/26 22:57, 2F
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07/26 22:58, , 3F
f(A) = 2*pi*r^2 + 2*pi*r*h
07/26 22:58, 3F
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07/26 22:59, , 4F
h = 500/pi/r^2
07/26 22:59, 4F
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07/26 22:59, , 5F
代入 f(A)
07/26 22:59, 5F
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07/26 22:59, , 6F
f(A) 就會變成 f(r)
07/26 22:59, 6F
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07/26 22:59, , 7F
若表面積要最小，至少一階導數要等於零
07/26 22:59, 7F
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07/26 23:00, , 8F
f'(r) = 0 , find r
07/26 23:00, 8F
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07/26 23:00, , 9F
then calculate h
07/26 23:00, 9F
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07/26 23:19, , 10F
thanks!
07/26 23:19, 10F
推
07/27 01:34, , 11F
CHECK f''(r)>0
07/27 01:34, 11F
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