1. Apr 4, 2012

### S.R

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. Apr 4, 2012

### HallsofIvy

Staff Emeritus
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 $h+ 2(l+w)\le 297$.

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 ($\lambda$ times the other.

3. Apr 4, 2012

### S.R

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

Last edited: Apr 5, 2012