Max Volume Parcel w/ Triangular Prism: Canada Post

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In summary, the problem involves finding the maximum volume of a triangular prism subject to certain constraints set by Canada Post. The boundaries are defined as (x+y+z+√(x^2+y^2)) ≤ 2 and x ≤ 1, and the function for volume is V(x,y,z) = 1/2xyz. The approach to solving this problem involves using the Karush-Kuhn-Tucker conditions or the Lagrangian method to find the critical points on the boundary, and then solving the system of equations to find the optimal values for x, y, and z. However, there may be an easier method to solve this problem.
  • #1
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Homework Statement


Canada Post accepts international parcels whose (Length+Girth) is less than or equal to 2 meters, and Length is less than or equal to 1 meter. Girth is defined as the cross section. We wish to ship a parcel of the shape of a triangular prism of length l meters. The cross section is a right triangle with catheti of lengths a and b meters. Assume the package walls are thin. What is the maximal volume of a parcel?


Homework Equations


Let
[tex]leg_1=a=y, leg_2=b=z, length=x[/tex]
I provided a drawing via paint for you to envision my take on the problem
Untitled-7.png

Hence,
[tex]V(x, y, z)=\frac{1}{2}xyz[/tex]
Boundaries:
[tex](x+y+z+\sqrt{x^2+y^2})\leq{2}, x\leq{1}[/tex]

The Attempt at a Solution


[tex]\nabla{f}(x, y, z)=\frac{1}{2}(yzi+xzj+xyk)[/tex]
and hence there exists a critical point at (0, 0, 0).

Next, I get partially lost. Should I be finding second partial derivatives of the boundary and then evaluating the Hessian matrix to determine extremes on the boundary?
 
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  • #2
dimpledur said:

Homework Statement


Canada Post accepts international parcels whose (Length+Girth) is less than or equal to 2 meters, and Length is less than or equal to 1 meter. Girth is defined as the cross section. We wish to ship a parcel of the shape of a triangular prism of length l meters. The cross section is a right triangle with catheti of lengths a and b meters. Assume the package walls are thin. What is the maximal volume of a parcel?


Homework Equations


Let
[tex]leg_1=a=y, leg_2=b=z, length=x[/tex]
I provided a drawing via paint for you to envision my take on the problem
Untitled-7.png

Hence,
[tex]V(x, y, z)=\frac{1}{2}xyz[/tex]
Boundaries:
[tex](x+y+z+\sqrt{x^2+y^2})\leq{2}, x\leq{1}[/tex]

The Attempt at a Solution


[tex]\nabla{f}(x, y, z)=\frac{1}{2}(yzi+xzj+xyk)[/tex]
and hence there exists a critical point at (0, 0, 0).

Next, I get partially lost. Should I be finding second partial derivatives of the boundary and then evaluating the Hessian matrix to determine extremes on the boundary?

Critical points of V are irrelevant in this problem, due to the presence of constraints. Here is a simpler example: what is the maximum of f(x) =x^2, subject to 0 <= x <= 1? Obviously, the max is at x = 1, but the derivative of f is 2 at that point, not zero.

RGV
 
  • #3
hmm, and when I tried finding critical points within the bounded portion the gradient told me there were not. It doesn't seem logical to just start plugging numbers in, as there are three variables that depend on girth.
 
  • #4
dimpledur said:
hmm, and when I tried finding critical points within the bounded portion the gradient told me there were not. It doesn't seem logical to just start plugging numbers in, as there are three variables that depend on girth.

You have forgotten three very important restrictions: x >= 0, y >= 0, z >= 0. If you were to submit your problem to a computer package, but without these restrictions, you would get a nonsensical solution: you can let x and y go to -infinity, and fix z = 1. The both of your written restrictions are satisfied, and V --> + infinity (which is certainly as much of a maximum as you could want!).

Anyway, your complete problem is:
max V(x,y,z)
subject to g(x,y,z) <= 2, x <= 1 and x, y, z >= 0.
(Here, V and g are the functions you wrote.)

You can solve this problem using the Karush-Kuhn-Tucker conditions (essentially, the Lagrangian conditions extended to inequality constraints). If all you want is a numerical solution, and don't care how you get it, you can use the EXCEL Solver tool to get a solution in an EXCEL spreadsheet. Other spreadsheets (including free, open source ones) also have solver tools.

Google 'Karush-Kuhn-Tucker' and 'Excel Solver' for more information.

RGV
 
  • #5
Okay, I have had a bit more experience with Lagrange multipliers, hence here is my next attempt.
The maximum must occur on the boundary of,
[tex](x+y+z+\sqrt{z^2+y^2})={2}[/tex]
Hence, let
[tex]L=0.5xyz+\lambda{(}x+y+z+\sqrt{y^2+z^2-2})[/tex]
The critical points of L are determined via
[tex]L_1=0.5yz+\lambda[/tex]
[tex]L_2=0.5xz+\lambda{+}\frac{2y}{\sqrt{y^2+z^2}}[/tex]
[tex]L_3=0.5xy+\lambda{+}\frac{2z}{\sqrt{y^2+z^2}}[/tex]
[tex]L_4=x+y+z+\sqrt{x^2+y^2}-2[/tex]
Is there any easy way to compute the solutions without using computational software?
 
  • #6
I guess the next step would be to solve the system of equations for the four unknowns, since there are four equations.

ALso was informed that there was an easier method to solving this question.
 
Last edited:

1. What is a Max Volume Parcel with Triangular Prism?

A Max Volume Parcel with Triangular Prism is a type of shipping box designed by Canada Post to maximize the volume of the package while minimizing the overall dimensions. It is shaped like a triangular prism, which allows for more efficient use of space while still conforming to Canada Post's maximum size and weight restrictions for parcels.

2. How much can a Max Volume Parcel with Triangular Prism hold?

The exact capacity of a Max Volume Parcel with Triangular Prism will vary depending on the specific dimensions chosen, but it is designed to hold up to 30% more volume than a traditional rectangular parcel of the same size. This means that you can fit more items into the parcel while still paying the same shipping rate.

3. Can I ship fragile items in a Max Volume Parcel with Triangular Prism?

Yes, you can ship fragile items in a Max Volume Parcel with Triangular Prism. However, it is recommended to use additional packaging materials, such as bubble wrap or packing peanuts, to provide extra protection for your items. Canada Post also offers insurance options for added peace of mind when shipping fragile items.

4. Is there an additional cost for using a Max Volume Parcel with Triangular Prism?

There is no additional cost for using a Max Volume Parcel with Triangular Prism. The price is determined by the weight and destination of the parcel, just like any other shipping method offered by Canada Post. However, because of the increased volume capacity, you may be able to save money by using this option for larger or heavier packages.

5. Can I track my Max Volume Parcel with Triangular Prism?

Yes, you can track your Max Volume Parcel with Triangular Prism using the tracking number provided by Canada Post. This allows you to monitor the progress of your package and ensure that it arrives at its destination on time. Tracking is included with all Canada Post shipping options, including the Max Volume Parcel with Triangular Prism.

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