GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A chef is experimenting with a new recipe. The cost of ingredients, C, in dollars, is given by the function C(x) = 0.5x + 3, where x is the number of servings. The revenue R, in dollars, from selling x servings is R(x) = 2x. At what number of servings will the cost equal the revenue (break-even point)?
- A4 servings
- B5 servings
- C3 servings
- D2 servings
Show answer & explanationAnswer & explanation
Correct answer: D. 2 servings
To find the break-even point, set the cost function equal to the revenue function: C(x) = R(x). So, 0.5x + 3 = 2x. Subtract 0.5x from both sides: 3 = 1.5x. Divide by 1.5: x = 3 / 1.5 = 2. The break-even point is 2 servings.
Why the other options are wrong
- A. At 4 servings, cost = 0.5(4)+3 = 5, revenue = 2(4) = 8. Not equal.
- B. At 5 servings, cost = 0.5(5)+3 = 5.5, revenue = 2(5) = 10. Not equal.
- C. At 3 servings, cost = 0.5(3)+3 = 4.5, revenue = 2(3) = 6. Not equal.
Solving Linear Systems (Break-Even)
A break-even point occurs when the cost of production equals the revenue generated from sales. Mathematically, it's the point where two functions (cost and revenue) intersect, found by setting their equations equal to each other.
- Cost function typically includes a fixed cost (y-intercept) and a variable cost (slope).
- Revenue function usually starts at zero and has a positive slope (price per item).
- The solution (x, y) represents the quantity (x) and the amount of money (y) at which break-even occurs.
Memory trick: When money in equals money out!