Patrick J.

asked • 09/15/18

Lagrange Multipliers

A parcel delivery service requires that the dimension of a rectangular box be such that the
length plus twice the width plus twice the height be no more than 108 centimetres. What is
the volume of the largest box that the company will deliver?

This uses inequality constraints. Any chance I could get some help on this? So far I know that I have to maximise the Volume V = lwh with the constraint L + 2w + 2h <= 108cm

1 Expert Answer

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Andrew R. answered • 09/15/18

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V=LWH and L+2W+2H≤108 so let g=L+2W+2H and ∇V=λ∇g
VL=WH,  VW=LH,  VH=LW   and   gL=1, gW=2,  gH=2  so
 
WH=λ, LH=2λ,  LW=2λ  --->  L=2W, L=2H,  H=W
so using g:  L+L+L≤108  L≤36,  2W+2W+2W ≤108  W≤18
and 2H+2H+2H≤108  H≤18
to maximize the volume let L=36  and  W=H=18
so g=L+2W+2H=36+2*18+2*18=108  and V=36*18*18
 
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Patrick J.

How about considering the constraint L+2W+2H<108 , because in my class we are taught to consider firstly L+2W+2H<108 and then L+2W+2H=108, so that all extrema found in both cases can be collated together as part of L+2W+2H≤108 and then analysed from there... but I ran into the problem of firstly setting ∇V = (0,0,0) because this gives l=w=h=0, and then also a problem with using the Hessian in three variables, because the hessian is greater than zero for any l, w and  h, but V_LL is zero and therefore I can't seem to conclude if (0,0,0) is a maximum, minimum or saddle point (for mathematical thoroughness; I knew from the start that (0,0,0) is definitely not the answer).
 
Thanks heaps for the help by the way, really appreciate it!
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09/15/18

Andrew R.

Why would you set  ∇V = (0,0,0)? You solve the resulting  eqs then find
L=W=H=0 is a solution, but this is clearly the min for V=LWH if
L>0, W>0 and H>0.
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09/20/18

Patrick J.

Yeah i think I was thinking of previous questions I'd done where it asked for the extrema (minima and maxima) but in this question it's obvious it wants maxima only, I was just wondering how i'd find the minima for this problem but it does seem like too much hassle if the question wasn't actually designed for it
 
Thanks for the help :)
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09/21/18

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