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 $\displaystyle x$ = 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.