Question

Use Lagrange multipliers to find the dimensions of a box WITHOUT A TOP of volume V=1 of minimal surface area.

Answer 1

Length * Height * Width= 1
Constraint -----> L H W =1 +G
minimize this --> L W + 2 W H + 2 LH =F

w.r.t <L, w, h>
Gradient F

<W+2H,L+2H, 2W+2L>

Gradient of G
<HW,LH,LW>

\[\lambda*HW=W+2H\]
\[\lambda*LH=L+2H\]
\[\lambda*LW=2L+2W\]

Answer 2

Thank you very, very much! Your work and response is extremely helpful!

Answer 3

\[\lambda*HW=W+2H\]
\[\lambda*LH=L+2H\]
\[\lambda*LW=2L+2W\]
\[L H W =1\]

Four equations, four variable