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Algebra 2 NYS Regents: Question 33 Explained

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Understanding the Algebra 2 NYS Regents Exam: Question 33 Explained

The Algebra 2 NYS Regents Exam is a significant milestone for high school students in New York State. It assesses their understanding of a wide range of algebraic concepts, including functions, equations, inequalities, and systems of equations. Question 33 on the exam is particularly challenging, often involving complex problem-solving scenarios. In this blog post, we'll delve into a typical Question 33, providing a step-by-step explanation to help you conquer this exam hurdle.

A Typical Question 33 Scenario

Let's consider a common type of Question 33 that involves a system of equations. The problem might present you with a scenario like this:

A company produces two types of products, A and B. Product A requires 3 hours of labor and 2 units of raw material per unit produced, while Product B requires 2 hours of labor and 4 units of raw material per unit. The company has a total of 120 hours of labor and 100 units of raw material available. The profit per unit of Product A is $5, and the profit per unit of Product B is $7. How many units of each product should the company produce to maximize its profit?

Breaking Down the Problem

This problem involves linear programming, a technique used to optimize a function (in this case, profit) subject to constraints (labor and raw material availability). Here's how to approach it:

1. Define Variables

  • Let x represent the number of units of Product A produced.
  • Let y represent the number of units of Product B produced.

2. Formulate Constraints

  • Labor Constraint: 3x + 2y ≤ 120 (The total labor hours used cannot exceed 120).
  • Raw Material Constraint: 2x + 4y ≤ 100 (The total raw material used cannot exceed 100).
  • Non-negativity Constraints: x ≥ 0 and y ≥ 0 (The company cannot produce a negative number of units).

3. Define the Objective Function

The objective function represents the profit the company wants to maximize. In this case, it's:

Profit (P) = 5x + 7y

4. Graph the Constraints

To visualize the feasible region (the area where all constraints are satisfied), we graph the constraints as inequalities on a coordinate plane. The feasible region is the area where all the inequalities overlap.

**Note:** You'll need to use the intercepts and slopes of each constraint to graph them accurately.

5. Find Corner Points

The corner points of the feasible region are the points where the constraint lines intersect. These points represent the potential solutions that maximize or minimize the objective function.

6. Evaluate the Objective Function at Corner Points

Substitute the coordinates of each corner point into the profit function (P = 5x + 7y). The corner point that yields the highest profit value is the optimal solution.

Solving the Problem

Let's illustrate this with an example. Suppose you find the following corner points for the feasible region:

  • (0, 0)
  • (40, 0)
  • (20, 20)
  • (0, 25)

Now, evaluate the profit function at each corner point:

  • (0, 0): P = 5(0) + 7(0) = $0
  • (40, 0): P = 5(40) + 7(0) = $200
  • (20, 20): P = 5(20) + 7(20) = $240
  • (0, 25): P = 5(0) + 7(25) = $175

The maximum profit of $240 is achieved when the company produces 20 units of Product A and 20 units of Product B.

Key Takeaways

Question 33 on the Algebra 2 NYS Regents Exam often involves linear programming. To solve these problems:

  • Define variables and formulate constraints based on the given information.
  • Graph the constraints to visualize the feasible region.
  • Identify the corner points of the feasible region.
  • Evaluate the objective function (profit, cost, etc.) at the corner points.
  • The corner point that yields the optimal value (maximum or minimum) is the solution.

Practice solving similar problems to gain confidence and mastery of this important concept for the Algebra 2 NYS Regents Exam.

NYS Regents Exam