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Algebra 2 NYS Regents Exam: Question 29 Explained
The Algebra 2 NYS Regents Exam is a challenging test that covers a wide range of topics. One of the most common types of questions on the exam is the word problem, which requires students to apply their algebraic knowledge to real-world scenarios. Question 29 is often a word problem that can be tricky for students to solve. In this blog post, we will provide a detailed explanation of a sample Question 29 from the Algebra 2 NYS Regents Exam and break down the steps involved in solving it.
Understanding the Problem
Here's a sample Question 29 from the Algebra 2 NYS Regents Exam:
A company produces a product that has a fixed cost of $1000 and a variable cost of $5 per unit. The company sells the product for $15 per unit. Let x represent the number of units produced and sold.
(a) Write a system of equations to represent the total cost, C(x), and the total revenue, R(x).
(b) Determine the break-even point for the company.
(c) Graph the cost and revenue functions on the same set of axes. Label the break-even point.
(d) Interpret the meaning of the break-even point in the context of the problem.
Step-by-Step Solution
(a) Writing the Cost and Revenue Equations
The total cost, C(x), is the sum of the fixed cost and the variable cost. The fixed cost is $1000, and the variable cost is $5 per unit, so the total cost is:
C(x) = 1000 + 5x
The total revenue, R(x), is the price per unit multiplied by the number of units sold. The price per unit is $15, so the total revenue is:
R(x) = 15x
(b) Determining the Break-Even Point
The break-even point is the point where the total cost equals the total revenue. In other words, it's the point where the company neither makes a profit nor incurs a loss. To find the break-even point, we set the cost and revenue equations equal to each other:
1000 + 5x = 15x
Solving for x, we get:
1000 = 10x
x = 100
Therefore, the break-even point is 100 units. This means that the company must produce and sell 100 units to cover all its costs.
(c) Graphing the Cost and Revenue Functions
To graph the cost and revenue functions, we can use a graphing calculator or plot points manually. Here is a sample graph:
The break-even point is where the cost and revenue lines intersect. This point is (100, 1500).
(d) Interpreting the Break-Even Point
The break-even point of 100 units means that the company needs to produce and sell at least 100 units to cover all its costs. If they produce and sell fewer than 100 units, they will incur a loss. If they produce and sell more than 100 units, they will make a profit.
Key Takeaways
Question 29 on the Algebra 2 NYS Regents Exam often involves word problems that require students to apply their algebraic knowledge to real-world scenarios. By understanding the concepts of cost, revenue, and break-even point, students can confidently solve these types of problems.
Remember to carefully read the problem, identify the key information, and use the appropriate algebraic techniques to solve it. Practice solving similar problems from past exams to improve your understanding and performance on the Algebra 2 Regents Exam.