What is the Smallest Surface Area of a Crate Delivered by Canada Post?

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SUMMARY

The discussion centers on calculating the smallest surface area (SA) of a crate that Canada Post will deliver, constrained by a maximum combined length and girth of 297 cm. The girth of a rectangular prism is defined as 2(l + w), where l is the length and w is the width. Participants suggest two methods for solving the problem: using partial derivatives to find critical points or applying the Lagrange Multiplier method to optimize the surface area function SA = 2(wh + lw + lh) under the given constraint.

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S.R
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Homework Statement


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?

Homework 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.

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:
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S.R said:

Homework Statement


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?
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?

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

The Attempt at a Solution


I set-up the equations:

l+2(w+h)=297
Hey, you switched h and l on me!

SA=2(wh+lw+lh)

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

S.R
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 (\lambda times the other.
 
HallsofIvy said:
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.

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

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