Daily Regents: Quadratic-Linear Systems of Equations

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 37

37. A company is considering building a manufacturing plant. They determine the weekly production cost at site A to be A(x) = 3x² while the production cost at site B is B(x) = 8x + 3, where x represents the number of products, in hundreds, and A(x) and B(x) are the production costs, in hundreds of dollars.

Graph the production cost functions on the set of axes below and label them site A and site B.

State the positive value(s) of x for which the production costs at the two sites are equal. Explain how you determined your answer.

If the company plans on manufacturing 200 products per week, which site should they use? Justify your answer.

It is confusing that everything is listed in hundreds, but it's okay, because they basically cancel out as far as the scale is concerned. Also a headache is that each box on the vertical side represents 2, so all the odd numbers are going to be between the lines.

You can put both of these functions into your graphing calculator and get the points you need.

If you don't know how to find them on the calculator, you can make two Tables of Values, as follows:

Plot each set of points. You can see my results below. (On my graph, I also plotted (.5, 1.5) so I could draw the curve better in my graph. You wouldn't be expected to.)

The two questions from the following page assume that you did the graph. However, they can be solved algebraically, without the graph. This is a bit more work, but it can be done. When they say to Explain your answer, you can refer to the graph, if you drew the graph. If you didn't, you have to show the computations you did to get the answer. Otherwise, it's assumed that you're just taking a guess. Any guess that's 50-50 (choice A or choice B) is NOT worth any credit.

The positive value of x is 3, which represents 300 units. (Note: x is NOT 300.)
They ask for the positive value, because if you solve algebraically, you may get a negative value as well, which would be discarded.

If only 200 units were made, the graph shows that site A is cheaper than site B.
Without a graph, you would have to calculate A(2) = 3(2)² = 12 and B(2) = 8(2) + 3 = 19 to determine that A is cheaper.

Any questions?

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If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.