Two-Variable Inequality
The two- variable linear inequality equation
Total amount of board feet of maple lumber available = 3000 board feet.
Amount of board feet of maple lumber to make one classic maple rocking chair=15 board feet
Amount of board feet of maple lumber to make one modern maple rocking chair=12 board feet
Assume the following
- The total number of classic maple rocking chairs to be made is x and
- The total number of modern maple rocking chairs to be made is
Then;
The total board feet of maple lumber required to make the classic maple rocking chairs x is 15*x=15x and
The total board feet of maple lumber required to make the modern maple rocking chairs is 12*y=12y
The inequality equation would be as follows;
If 3000 board feet of maple lumber is available then the total board feet of maple lumber to make x classic rocking chairs is 3000-15x.
The remaining board feet of maple lumber would be the 12y for making the modern maple rocking chairs. Hence the linear inequality equation
3000-15x<= 12y
Solve for y;
12y/12<=3000-15x/12, y<=3000/12-15x/12,
y<=250-5/4x
Plotting the graph of inequality
Assume the following values of x for the y values below;
| x | Y<=250-5/4x |
| 0 | 250 |
| 20 | 225 |
| 40 | 200 |
| 60 | 175 |
| 80 | 150 |
| 100 | 125 |
| 120 | 100 |
| 140 | 75 |
| 160 | 50 |
| 180 | 25 |
| 200 | 0 |
| 220 | -25 |
| 240 | -50 |
The graph is linear and intersects once and that is when the total number of modern chairs y to be made is 111 and the classic chairs are 111.The shaded area to fulfill the inequality is marked by figures for x- axis above each value of x in the table above and figures below each value of y in the table above. The area between values 0 to 100 for y axis and 125 to 250 for x axis is the area not shaded and is the one that fulfils the inequality.
The application of the findings in the graph are that when Ozark Furniture Company, for example makes 200 classic maple chairs then it will not have enough board feet of maple lumber left to make the modern rocking maple chairs. If the company makes125 modern maple chairs then the board feet of maple lumber left will be enough to make 100 classic chairs only. The number of each rocker made is depended on the number of the other rocker to be made as shown in the table above. If the company decides for example to make 220 classic maple chairs then it will not have enough board feet of maple lumber to make the 220 classic rockers and it will need to buy additional board feet of maple lumber to complete making the classic rockers and will have to buy additional board feet of maple lumber for any modern rocker they have to make.
Analysis of order from Chain Store
The 3000 board feet of maple lumber can only be used to make 125 classic maple rocking chairs and 93 modern maple chairs. This is as shown below;
Y=250-5/4x if x=250 the y would be 250-5/4*125=93.75
The 93 chairs are the number of modern rocking chairs that can be made.
The company cannot therefore, supply this order with the 3000 board feet of maple lumber. The company will therefore need to add the following amount of board feet of maple lumber to make the remaining modern maple chairs.
Remaining modern maple rocking chairs —175-93=82.The board feet of maple lumber to make one modern rocker is 12. To make 82 modern rockers the company will need to purchase 984 board feet of maple lumber calculated as follows;
82modern rockers*12 board feet of maple lumber=984 board feet of maple lumber
The company will therefore need a total of 3,984 board feet of marble lumber to supply the order of 175 modern rocking chairs and 125 classic rocking chairs.
References
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