Question

He earns $25 per hour for training and $15 per hour for data entry support. Steven can work no more

than 40 hours a week. He wants to use linear programming techniques to optimize his income.
Part A
Construct a model to represent the amount of time Steven can spend on data entry and training. Be sure
to define your variables. Sketch a graph of your model.
Part B
While Steven would like to spend most of his time training, the number of hours spent training cannot
exceed twice the number of hours doing data entry support. Steven must spend at least 10 hours each
week on data support. Construct a model to show the constraints. Sketch a graph of your model.
Part C
Find the vertices of the feasible region. Show your work or explain how you found the coordinates.
Part D
What is the maximum possible amount of money Steven can earn in a week given the constraints? Round
to the nearest dollar. Validate your solution to assess the reasonableness of you
Part E
The number of hours Steven worked one week resulted in a gross income of $800. From this, a portion was
withheld for benefits, retirement, and taxes. The total amount withheld from Steven’s check was $264.
The amount withheld for taxes was twice the amount withheld for retirement, and the amount withheld
for benefits was $24 less than the sum of retirement and taxes. Construct a system of equations that can
be used to find the amount of benefits, retirement, and taxes. Be sure to define your variables.
Part F
Solve the system from Part E. Validate your solution to assess the reasonableness of your model.

Answer 1

In Part A, the variables x and y represent the amount of time Steven can spend on data entry and training. The graph of this model would have x on the horizontal axis and y on the vertical axis. In Part B, the constraints are that the number of hours spent training cannot exceed twice the number of hours spent on data entry, and Steven must spend at least 10 hours per week on data entry.

Step-by-step explanation:

Part A

The variables that can be used to represent the amount of time Steven can spend on data entry and training are:
- Let x be the number of hours spent on data entry.
- Let y be the number of hours spent on training.

The graph of this model would have x on the horizontal axis and y on the vertical axis. The feasible region would be the area where x and y are both non-negative and satisfy the given constraints.

Part B

The constraints for Steven's situation are:
- The number of hours spent training cannot exceed twice the number of hours spent on data entry: y ≤ 2x.
- Steven must spend at least 10 hours per week on data entry: x ≥ 10.

The graph of this model would show the feasible region where the constraints are satisfied, which would be the area below the line y = 2x and to the right of the line x = 10.