Matrices: Problem

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October 9th 2012, 09:44 PM
seananigans

Matrices: Problem

Hello All,

I have been having much difficulty understanding matrices and a group of us have become desperate to find some comprehension of this problem. We have come up with an answer for c, but still not sure if it's correct. If you could please be of assistance it would be greatly appreciated!

A company makes three types of product (A,B, and C) with specifications below:

Product | A | B | C | Total Allowed
Weight 20 20 60 220 lbs
Volume 20 15 20 125 cu ft

a) Write a system of equations that could be used to determine what combination of products should be shipped.

b) Write the augmented matrix that matches the system.

c) Perform row operations to obtain a reduced matrix.

d) Interpret the matrix to determine the quantities to be shipped.
October 10th 2012, 12:15 AM
chiro

Re: Matrices: Problem

Hey seananigans.

Are there constraints for the total volume and the total weight?
October 10th 2012, 05:31 AM
seananigans

Re: Matrices: Problem

Hello Chiro,

I believe it would be as follows

Total Allowed Weight: 220lbs

Total Allowed Volume: 125 cu ft
October 10th 2012, 09:59 AM
Soroban

Re: Matrices: Problem

Hello, seananigans1

Quote:

A company makes three types of product (A,B, and C) with specifications below:

. . $\begin{array}{|c||c|c|c||c|} \hline \text{Product} & A & B & C & \text{Maximum} \\ \hline \text{Weight} & 20 & 20 & 60 & 220\text{ lbs} \\ \text{Volume} & 20 & 15 & 20 & 125\text{ cu ft} \\ \hline\end{array}$

a) Write a system of equations that could be used
. . to determine what combination of products should be shipped.

$\text{Let }\:\begin{Bmatrix} A &=& \text{units of A to be shipped} \\ B &=& \text{units of B to be shipped} \\ C &=& \text{units of C to be shipped} \end{Bmatrix} \quad A,B,C\:\ge\:0$

Then we have: . $\begin{Bmatrix}20A + 20B + 60C \:\le\:220 & \Rightarrow & A + B + 3C \:\le\:11 \\ 20A + 15B + 20C \:\le\:125 & \Rightarrow & 4A + 3B + 4C \:\le\:25 \end{Bmatrix}$

Quote:

b) Write the augmented matrix that matches the system.

$\left|\begin{array}{ccc|c} 1&1&3&11 \\ 4&3&4&25 \end{array}\right|$

Quote:

c) Perform row operations to obtain a reduced matrix.

We have: . $\left|\begin{array}{ccc|c} 1&1&3&11 \\ 4&3&4&25 \end{array}\right|$

$\begin{array}{c} \\ R_2-4R_1\end{array}\:\left|\begin{array}{ccc|c}1&1&3&1 1 \\ 0&\text{-}1&\text{-}8&\text{-}19 \end{array}\right|$

$\begin{array}{c}R_1+R_2 \\ \text{-}1R_2 \end{array}\:\left|\begin{array}{ccc|c} 1&0&\text{-}5& \text{-}8 \\ 0&1&8&19 \end{array}\right|$

Quote:

d) Interpret the matrix to determine the quantities to be shipped.

We have: . $\begin{Bmatrix}A -5C \:=\:\text{-}8 & \Rightarrow & A &=& 5C - 8 \\ B+8C \:=\:19 & \Rightarrow & B &=& \text{-}8C + 19 \\ && C &=& C\end{Bmatrix}$

$\text{On the right, replace }C\text{ with a parameter }t:\;\begin{Bmatrix} A &=& 5t - 8 \\ B &=& \text{-}8t+19 \\ C &=& t \end{Bmatrix}$

We have parametric equations for $A,B,C$ for any value of $t.$

But $A,B,C$ must be nonnegative.

$A\,\ge\,0 \quad\Rightarrow\quad 5t-8\:\ge\:0 \quad\Rightarrow\quad t \:\ge\:\tfrac{8}{5}$

$B\,\ge\,0 \quad\Rightarrow\quad \text{-}8t + 19 \:\ge\:0 \quad\Rightarrow\quad t \:\le\;\tfrac{19}{8}$

We have: . $1\tfrac{3}{5}\:\le\:t\:\le\:2\tfrac{3}{8} \quad\Rightarrow\quad t \,=\,2$

Therefore: . $\begin{Bmatrix} A&=& 2 \\ B&=&3 \\ C&=&2 \end{Bmatrix}$
October 10th 2012, 04:09 PM
chiro

Re: Matrices: Problem

Sorry, I was thinking this was some sort of operations research problem where you had packing constraints (like a trucks storage or something along those lines)
October 10th 2012, 08:36 PM
seananigans

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Re: Matrices: Problem

Thank you very much Soroban! This is a load of pressure off my chest, as well as our groups - I do have a question regarding both part A, if you don't mind.
I'm having difficulty understanding why and how the initial matrix is rewritten as is in part A.

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