I'm having trouble setting up the variables and constraints in these questions. Wondering if someone can help? Thanks!

http://img216.imageshack.us/img216/6991/2zq1.jpg

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- Aug 11th 2006, 08:50 AMclassicstringsApplications of Linear Programming
I'm having trouble setting up the variables and constraints in these questions. Wondering if someone can help? Thanks!

http://img216.imageshack.us/img216/6991/2zq1.jpg - Aug 11th 2006, 08:36 PMSoroban
Hello, classicstrings!

Here's the second one . . .

Quote:

A furniture maker produces cupboards and bookshelves.

Both products use wooden planks, sawing time, sanding time, and assembly time.

The cupboards use 20 m of wood, 40 minutes of sawing, 60 minutes of sanding,

and 10 minute of assembly time for a profit of $280.

The bookshelves use 10 m of wood, 30 minutes of sawing, 90 minutes of sanding,

and 30 minutes of assembly time for a profit of $340.

There are 220 m of wood, 480 minutes of sawing time, 1080 minutes of sanding time,

and 330 minutes of assembly time available.

How many of each item should be made for maximum profit?

Let $\displaystyle x$ = number of cupboards, $\displaystyle y$ = number of bookshelves. .$\displaystyle x \geq 0,\;y \geq 0$**[1]**

A chart helps to organize the information . . .Code:`| wood + saw | sand | assem |`

- - - - - - - - + - - -+ - - + - - -+ - - - +

cupboards (x) | 20x | 40x | 60x | 10x |

- - - - - - - - + - - + - - + - - -+ - - - +

bookshelves (y) | 10y | 30y | 90y | 30y |

- - - - - - - - + - - + - - + - - -+ - - - +

available | 220 | 480 | 1080 | 330 |

- - - - - - - - + - - -+ - - + - - -+ - - - +

Wood: .$\displaystyle 20x + 10y \:\leq \:220\quad\Rightarrow\quad 2x + y \:\leq \:22$**[2]**

Sawing: .$\displaystyle 40x + 30y \:\leq \:480\quad\Rightarrow\quad 4x + 3y \:\leq \:48$**[3]**

Sanding: .$\displaystyle 60x + 90y \:\leq \:1080\quad\Rightarrow\quad 2x + 3y \:\leq \:36$**[4]**

Assembly: .$\displaystyle 10x + 30y \:\leq \:330\quad\Rightarrow\quad x + 3y\:\leq\:33$**[5]**

**[1]**places us in Quadrant 1.

**[2]**Graph the__line__: $\displaystyle 2x + y \:=\:22$. .It has intercepts: $\displaystyle (11,0),\;(0,22)$. .

. . .Shade the region below the line.

**[3]**Graph the line: $\displaystyle 4x + 3y \:=\:48$. .It has intercepts: $\displaystyle (12,0),\;(0,16)$

. . .Shade the region below the line.

**[4]**Graph the line: $\displaystyle 2x + 3y \:=\:36$. .It has intercepts: $\displaystyle (18,0),\;(0,12)$

. . .Shade the region below the line.

**[5]**Graph the line: $\displaystyle x + 3y \:=\:33$. .It has intercepts: $\displaystyle (33,0),\;(0,11)$

. . .Shade the region below the line.

The final region is a hexagon. .Its vertices are (clockwise from the origin):

. . $\displaystyle (0,0),\;(0,11),\;(3,10),\;)6,8),\;(9,4),\;(0,11)$

Test them in the profit function: $\displaystyle P \:= \:280x + 340y$

. . to see which one produces maximum profit.

- Aug 13th 2006, 01:09 AMclassicstrings
Hey Soroban! You have done the harder one for me, and I have gone through it a couple of times myself after, I have done the first one by looking @ how you did them. Cheers!