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Quadratic Properties Question

  1. Apr 4, 2012 #1

    S.R

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    1. The problem statement, all variables and given/known data
    Canada Post will deliver parcels only if they are less than a certain maximum size: the combined length and girth cannot exceed 297 cm (Girth is the total distance around the cross-section of the parcel). Canada Post delivers a crate with the smallest SA to your house. What is the SA of the crate in square meters?

    2. Relevant equations
    Girth of a rectangular prism=2(w+h) -> Web-search
    I'm still unclear on what a girth is, however. Maybe it is 2(l+w), the perimeter of the base.

    3. The attempt at a solution
    I set-up the equations:

    l+2(w+h)=297

    SA=2(wh+lw+lh)

    I'm unsure how to proceed. Any help is appreciated. Thanks!

    S.R
     
    Last edited: Apr 4, 2012
  2. jcsd
  3. Apr 4, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The smallest? You say that Canada Post will not deliver packages above a certain size but they can be as small as you please. Are you asking for the smallest surface are of a package that meets the maximum sum of length and girth?

    Yes. assuming you are taking h as the longest side, "girth" is 2(l+ w) so the requirement is that [itex]h+ 2(l+w)\le 297[/itex].

    Hey, you switched h and l on me!

    Proceed in either of two ways:
    1) use l+ 2*(w+ h)= 297 (or h+ 2(w+ l)= 297) to eliminate one of the three variables leaving only two. Set the partial derivatives with respect to the two variables equal to 0 and solve the two equations.

    2) Use the "Lagrange Multiplier" method. Form the gradient of the "object function", 2(wh+ lw+ lh), the gradient of the constraint, l+ 2(w+h), and set one equal to a constant ([itex]\lambda[/itex] times the other.
     
  4. Apr 4, 2012 #3

    S.R

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    I'm not sure how to eliminate one of the three variables?
     
    Last edited: Apr 5, 2012
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