January 2024 Algebra 2 Regents Part IV

This exam was adminstered in January 2024.

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Algebra 2 January 2024

Part IV: A correct answer will receive 6 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.


37.A manufacturer of sweatshirts finds that profits and costs fluctuate depending on the number of products created. Creating more products doesn’t always increase profits because it requires additional costs, such as building a larger facility or hiring more workers. The manufacturer determines the profit, p(x), in thousands of dollars, as a function of the number of sweatshirts sold, x, in thousands. This function, p, is given below.

p(x) = -x³ + 11x² - 7x - 69

Graph y = p(x), over the interval 0 < x < 9, on the set of axes below

Over the given interval, state the coordinates of the maximum of p and round all values to the nearest integer. Explain what this point represents in terms of the number of sweatshirts sold and profit.

Determine how many sweatshirts, to the nearest whole sweatshirt, the manufacturer would need to produce in order to first make a positive profit. Justify your answer.

Answer:

Put the function in your graphing calculator and get the Table of Values. Plot the points on the graph from 0 through 9. Connect them but DO NOT add any arrows that go beyond the specified range. Notice that each box on the y-axis is 15 units, while each box on the x-axis is one.

The points you need to plot are (0,-69), (1,-66), (2,-47), (3,-18), (4,15), (5,46), (6,69), (7,78), (8,67), (9,30).

Your graph will look like this:

The coordinates maximum value, to the nearest interger, are (7,78). This means that for 7,000 sweatshirts the profit will be $78,000.

Using the calculator, you can find the zero of the function to be (3.5488,0). Since the function represents thousands of sweatshirts, then this is 3548.8 sweatshirts. That means that the first positive profit will be at 3549 sweatshirts.

End of Exam

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