# Math Help - Applications of Linear Programming

1. ## Applications of Linear Programming

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

2. Hello, classicstrings!

Here's the second one . . .

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 $x$ = number of cupboards, $y$ = number of bookshelves. . $x \geq 0,\;y \geq 0$ [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: . $20x + 10y \:\leq \:220\quad\Rightarrow\quad 2x + y \:\leq \:22$ [2]

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

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

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

[1] places us in Quadrant 1.

[2] Graph the line: $2x + y \:=\:22$. .It has intercepts: $(11,0),\;(0,22)$. .
. . .Shade the region below the line.

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

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

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

The final region is a hexagon. .Its vertices are (clockwise from the origin):
. . $(0,0),\;(0,11),\;(3,10),\6,8),\;(9,4),\;(0,11)" alt="(0,0),\;(0,11),\;(3,10),\6,8),\;(9,4),\;(0,11)" />

Test them in the profit function: $P \:= \:280x + 340y$
. . to see which one produces maximum profit.

3. 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!