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 = number of cupboards, = number of bookshelves. . [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: . [2]
Sawing: . [3]
Sanding: . [4]
Assembly: . [5]
[1] places us in Quadrant 1.
[2] Graph the line: . .It has intercepts: . .
. . .Shade the region below the line.
[3] Graph the line: . .It has intercepts:
. . .Shade the region below the line.
[4] Graph the line: . .It has intercepts:
. . .Shade the region below the line.
[5] Graph the line: . .It has intercepts:
. . .Shade the region below the line.
The final region is a hexagon. .Its vertices are (clockwise from the origin):
. . 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:
. . to see which one produces maximum profit.