Our box has 6 sides. Obviously, front and back sides are the same. So are top and bottom sides. And left and right sides too.
We can calculate areas of each side. And then multiplying it by its cost we can find its total price.
For example, dimentions of the front side are w w w h h h S 1 = w h S_1=wh S 1 = w h P 1 = 3 w h P_1=3wh P 1 = 3 w h
f ( w , l , h ) = 6 w h + 6 l h + 8 w l f(w,l,h)=6wh+6lh+8wl f ( w , l , h ) = 6 w h + 6 l h + 8 wl
Obviously, w l h = 452 wlh=452 wl h = 452
We should minimize f f f w l h = 452 wlh=452 wl h = 452
Let's construct Lagrange function.
L ( w , l , h ) = 6 w h + 6 l h + 8 w l + λ ( w l h − 452 ) L(w,l,h)=6wh+6lh+8wl+\lambda (wlh-452) L ( w , l , h ) = 6 w h + 6 l h + 8 wl + λ ( wl h − 452 )
Derivatives are:
∂ L ∂ w = 6 h + 8 l + λ l h \frac{\partial L}{\partial w}=6h+8l+\lambda lh ∂ w ∂ L = 6 h + 8 l + λ l h
∂ L ∂ l = 6 h + 8 w + λ w h \frac{\partial L}{\partial l}=6h+8w+\lambda wh ∂ l ∂ L = 6 h + 8 w + λ w h
∂ L ∂ h = 6 w + 6 l + λ w l \frac{\partial L}{\partial h}=6w+6l+\lambda wl ∂ h ∂ L = 6 w + 6 l + λ wl
Next step is to solve the system
{ 6 h + 8 l + λ l h = 0 6 h + 8 w + λ w h = 0 6 w + 6 l + λ w l = 0 w l h = 452 \begin {cases}
6h+8l+\lambda lh=0\\
6h+8w+\lambda wh=0\\
6w+6l+\lambda wl = 0\\
wlh=452
\end {cases} ⎩ ⎨ ⎧ 6 h + 8 l + λ l h = 0 6 h + 8 w + λ w h = 0 6 w + 6 l + λ wl = 0 wl h = 452 ⟺ \iff ⟺ { h ( 6 + λ l ) = − 8 l 6 h + 8 w + λ w h = 0 w ( 6 + λ l ) = − 6 l w l h = 452 \begin {cases}
h(6+\lambda l)=-8l\\
6h+8w+\lambda wh=0\\
w(6+\lambda l)=-6l\\
wlh=452
\end {cases} ⎩ ⎨ ⎧ h ( 6 + λ l ) = − 8 l 6 h + 8 w + λ w h = 0 w ( 6 + λ l ) = − 6 l wl h = 452 ⟺ \iff ⟺ { h = − 8 l ( 6 + λ l ) 6 h + 8 w + λ w h = 0 w = − 6 l ( 6 + λ l ) w l h = 452 \begin {cases}
h=\frac{-8l}{(6+\lambda l)}\\
6h+8w+\lambda wh=0\\
w=\frac {-6l}{(6+\lambda l)}\\
wlh=452
\end {cases} ⎩ ⎨ ⎧ h = ( 6 + λ l ) − 8 l 6 h + 8 w + λ w h = 0 w = ( 6 + λ l ) − 6 l wl h = 452 ⟺ \iff ⟺ { h = − 8 l ( 6 + λ l ) 6 − 8 l ( 6 + λ l ) + 8 − 6 l ( 6 + λ l ) + λ − 8 l ( 6 + λ l ) − 6 l ( 6 + λ l ) = 0 w = − 6 l ( 6 + λ l ) − 6 l ( 6 + λ l ) l − 8 l ( 6 + λ l ) = 452 \begin {cases}
h=\frac{-8l}{(6+\lambda l)}\\
6\frac{-8l}{(6+\lambda l)}+8\frac {-6l}{(6+\lambda l)}+\lambda \frac{-8l}{(6+\lambda l)}\frac {-6l}{(6+\lambda l)}=0\\
w=\frac {-6l}{(6+\lambda l)}\\
\frac {-6l}{(6+\lambda l)}l\frac{-8l}{(6+\lambda l)}=452
\end {cases} ⎩ ⎨ ⎧ h = ( 6 + λ l ) − 8 l 6 ( 6 + λ l ) − 8 l + 8 ( 6 + λ l ) − 6 l + λ ( 6 + λ l ) − 8 l ( 6 + λ l ) − 6 l = 0 w = ( 6 + λ l ) − 6 l ( 6 + λ l ) − 6 l l ( 6 + λ l ) − 8 l = 452 ⟺ \iff ⟺ { h = − 8 l ( 6 + λ l ) 2 l ( 6 + λ l ) = λ l 2 ( 6 + λ l ) 2 w = − 6 l ( 6 + λ l ) 48 l 3 ( 6 + λ l ) 2 = 452 \begin {cases}
h=\frac{-8l}{(6+\lambda l)}\\
\frac{2l}{(6+\lambda l)}= \frac{\lambda l^2}{(6+\lambda l)^2}\\
w=\frac {-6l}{(6+\lambda l)}\\
\frac {48l^3}{(6+\lambda l)^2}=452
\end {cases} ⎩ ⎨ ⎧ h = ( 6 + λ l ) − 8 l ( 6 + λ l ) 2 l = ( 6 + λ l ) 2 λ l 2 w = ( 6 + λ l ) − 6 l ( 6 + λ l ) 2 48 l 3 = 452 ⟺ \iff ⟺ { h = − 8 l ( 6 + λ l ) 2 l ( 6 + λ l ) = λ l 2 w = − 6 l ( 6 + λ l ) 48 l 3 ( 6 + λ l ) 2 = 452 \begin {cases}
h=\frac{-8l}{(6+\lambda l)}\\
2l(6+\lambda l)= \lambda l^2\\
w=\frac {-6l}{(6+\lambda l)}\\
\frac {48l^3}{(6+\lambda l)^2}=452
\end {cases} ⎩ ⎨ ⎧ h = ( 6 + λ l ) − 8 l 2 l ( 6 + λ l ) = λ l 2 w = ( 6 + λ l ) − 6 l ( 6 + λ l ) 2 48 l 3 = 452 ⟺ \iff ⟺ { h = − 8 l ( 6 + λ l ) λ l = − 12 w = − 6 l ( 6 + λ l ) 48 l 3 ( 6 + λ l ) 2 = 452 \begin {cases}
h=\frac{-8l}{(6+\lambda l)}\\
\lambda l=-12\\
w=\frac {-6l}{(6+\lambda l)}\\
\frac {48l^3}{(6+\lambda l)^2}=452
\end {cases} ⎩ ⎨ ⎧ h = ( 6 + λ l ) − 8 l λ l = − 12 w = ( 6 + λ l ) − 6 l ( 6 + λ l ) 2 48 l 3 = 452 ⟺ \iff ⟺ { h = 4 l 3 λ l = − 12 w = l 48 l 3 36 = 452 \begin {cases}
h=\frac {4l}{3}\\
\lambda l=-12\\
w=l\\
\frac {48l^3}{36}=452
\end {cases} ⎩ ⎨ ⎧ h = 3 4 l λ l = − 12 w = l 36 48 l 3 = 452
From the last equation it follows l 3 = 339 l^3=339 l 3 = 339 l = 339 3 l=\sqrt[3]{339} l = 3 339
{ h = 4 339 3 3 λ = − 12 339 3 w = 339 3 l = 339 3 \begin {cases}
h=\frac {4\sqrt[3]{339}}{3}\\
\lambda = \frac {-12}{\sqrt[3]{339}}\\
w=\sqrt[3]{339}\\
l=\sqrt[3]{339}
\end {cases} ⎩ ⎨ ⎧ h = 3 4 3 339 λ = 3 339 − 12 w = 3 339 l = 3 339
#333702 on Dec 2022
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