Re: [多變]
※ 引述《iips (朝目標前進中)》之銘言：
: Find the dimensions that will minimize the surface
: area (and hence the cost) of a rectangular fish aquarium,
: open on top,with a volume of 32 ft^3.
: Ans:4 ft by 4ft for the base;2 ft for the height
令 f(x,y,z) = xy + 2xz + 2yz , g(x,y,z) = xyz - 32
F(x,y,z) = f(x,y,z) + (λ)(g(x,y,z))
= (xy + 2xz + 2yz) + (λ)(xyz - 32)
Fx = y + 2z + (λ)(yz) = 0
=> -y - 2z = (λ)(yz)
-y - 2z
=> λ = --------- ------(1)
yz
Fy = x + 2z + (λ)(xz) = 0
=> -x - 2z = (λ)(xz)
-x - 2z
=> λ = --------- ------(2)
xz
Fz = 2x + 2y + (λ)(xy) = 0
=> -2x - 2y = (λ)(xy)
-2x - 2y
=> λ = ---------- ------(3)
xy
(1) x y + 2z
----- => (---)(--------) = 1
(2) y x + 2z
=> (x)(y + 2z) = (y)(x + 2z)
=> xy + 2xz = xy + 2yz => xz = yz => x = y (z > 0) ------(4)
(2) y x + 2z
----- => (---)(---------) = 1
(3) z 2x + 2y
=> (y)(x + 2z) = (z)(2x + 2y)
=> xy + 2yz = 2xz + 2yz => xy = 2xz => y = 2z (x > 0) ------(5)
由(4)、(5)得
x = 2z , y = 2z 代入 xyz = 32 得
(2z)(2z)(z) = 32 => z^3 = 8 => z = 2
x = 2z = 4 , y = 2z = 4
因此當 x = 4 , y = 4 , z = 2 時
f(x,y,z)有最小值
f(4,4,2) = (4)(4) + (2)(4)(2) + (2)(4)(2)
= 16 + 16 + 16 = 48
因此最小表面積為 48
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