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

[S.R]

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

 

[HallsofIvy]

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.

 

[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.

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