John looks for maximizing his weekly earnings by sharing 20 hours per week to work both as a bookkeeper and as a tutor. He earns $15 per hour as a bookkeeper and $16 per hour for tutoring. However, the tutoring center requires that he spends at least 4 hours but no more than 12 hours per week tutoring. John is convinced that he can maximize his earnings by working for 11 hours as a bookkeeper plus 9 hours as a tutor.
We will use the Linear Programming Theory to check if that would be the best choice for him or, if not, to indicate the one.
xxxx (Explain the Linear Programming Theory) xxxxxxx
In order to solve this problem, we first need to write the objective function that describes John´s total weekly earnings.
z = 15x + 16y, representing:
'z' - total earnings
'15x' - $15/hour times 'x' number of hours working as a bookkeeper
'16y' - $16/hour times 'y' number of hours working as a tutor
Next step, we need to write an inequality for each constraint that John is bound, as follow:
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Constraint
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Inequality
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Can´t work more than 20 hours per week
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x + y < 20
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Must work at least 4 hours per week tutoring
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x > 4
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Can´t work more than 12 hours per week tutoring
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x < 12
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Developing each inequality in order to graph the system, but considering only the first quadrant since 'x' and 'y' are non-negative:
x + y < 20
For 'x' = 0; y = 20 and for 'y' = 0, x = 20. So: (0,20) and (20,0), as a solid line.
Using (0,0) as a test point, 0+0 < 20 is True, the inequality is satisfied and therefore the graph should be shade below and to the left of the line.
x > 4
In that case we consider x = 4, as a solid line on the graph.
Using (0,0) as a test point, 0 > 4 is False, the inequality is not satisfied and therefore the graph should be shade to the right of the line.
xxxx (plot a graph) xxxxx
x < 12
Again, considering x = 12, as a solid line on the graph.
Using (0,0) as a test point, 0 < 12 is True, the inequality is satisfied and therefore the graph should be shade to the left of the line.
xxxx (plot a graph) xxxxx
Finally, the graph of the system of inequalities uses only the first quadrant because x and y are nonnegative, and is represented below:
xxxx (plot the final graph) xxxxx
By the graph above, its four vertices occur at (4,0) , (12,0) , (4,16) and (12,8)
The following table evaluates the objective function for total weekly earnings at each of the four vertices:
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VERTICE
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z = 15x + 16y
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(4,0)
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z = 15 (4) = 60
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(12,0)
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z = 15 (12) = 180
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(4,16)
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z = 15(4) + 16(16) = 316
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(12,8)
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z = 15 (12) + 16 (8) = 308
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Based on that, John can earn the maximum amount per week by working as a bookkeeper for 4 hours per week, earning $15 per hour, and tutoring for 16 hours per week, earning $16 per hour. The maximum amount that he can earn each week is $316.00.
John is wrong in his claim that he will maximize his weekly earnings by working for 11 hours as a bookkeeper and for 9 hours as a tutor, since he would earn less: $309.00 per week.
Personal Development, a Saint Leo core value, has been considered in this study once every person should look for a balanced life. It means that John, by earning the maximum possible on his available working time, will have more money to pay his bills or for pleasure. In other words, a better life at this point of his life.
In conclusion, Linear Programming Theory has been applied in daily situations as John´s dilemma. By using formulas that represents scenarios and its constraints, Linear Programming helps us to find the optimum choice. John was about to make a mistake by choosing an option that would make him lose $7.00 per week. Indeed he can earn the maximum amount, $316.00 per week, by working as a bookkeeper for 4 hours and tutoring for 16 per week.