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Graphing inequalities with calculator to find the maximum earnings of two part-time jobs

Q: Anna is a college student with two part-time jobs, one at a hardware store and the other at a fitness centre. She wants to work a minimum of 7 h/week at the hardware store, and 5h/week at the fitness centre . She does not want to work more than 15 h/ week. the hardware store pays $15/h, and the fitness centre pays $13/H. How many hours should she work at each job each week to maximize her earnings?

A: Let x = hours worked at hardware store
Let y = hours worked at fitness center

For these types of questions, you need to figure out what the constraints or restrictions are are. Constraints are the restrictions or limiting factors, labelled in blue text “Constraints” on the attached image. Notice that the last constraint is what determines the sloped line from y=15 down to x=15.

The TI-84 Plus C Silver Edition. It’s really easy to set up a system of inequalities on a graphing calculator and visualize the regions. It’s a great tool to have to supplement and verify the solutions from your hand drawn sketches.

Maximum earnings inequality graph

 
I have mixed feelings about graphing by hand versus using technology. In this case, graphing inequalities with a calculator is pretty simple, though it does require some knowledge of manipulating the symbols at the start of each line in the y= input menu.

  • At the x-intercept of the line (15,0) if she works 15 hours at the hardware store, she would have to work zero hours at the fitness center
  • At the y-intercept of the line (0,15), it’d be 15 hours at the fitness center, and zero hours at the hardware store. The intercepts represent the extremes.

The other two constraints, shown by red and green lines, restrict your area to the little triangle that I shaded in orange.

You then have to use the revenue or earnings equation to figure out where in that triangle the max earnings will be. Check the vertices of the triangle first.

The first guess would be to work the most hours possible at the hardware store, so x=10, using the equation for earnings:

E = 15x + 13y
E = 15(10) + 13(5)
E = $215

If you go up to the top of the triangle, where x=7:

E = 15(7) + 13(8)
E = $209

You can try a few more points, but I think you will find that the max earnings occur when x=10, y=5, giving an earnings of $215.