GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A financial analyst is modeling the profit (P) of a company, which is described by the equation P = -2x^2 + 24x - 50, where 'x' is the number of units sold (in thousands). What is the maximum profit the company can achieve?
- A$22,000
- B$50,000
- C$94,000
- D$72,000
Show answer & explanationAnswer & explanation
Correct answer: A. $22,000
Since the leading coefficient (-2) is negative, the parabola opens downwards, indicating a maximum profit. The x-coordinate of the vertex (number of units for max profit) is x = -b/(2a) = -24/(2*(-2)) = -24/-4 = 6. Substitute x=6 into the profit equation: P = -2(6)^2 + 24(6) - 50 = -2(36) + 144 - 50 = -72 + 144 - 50 = 72 - 50 = 22. Since x is in thousands, the profit is $22,000.
Why the other options are wrong
- B. This is the y-intercept, which represents the initial loss or fixed cost, not the maximum profit.
- C. This is an incorrect calculation, possibly from misinterpreting parts of the quadratic formula or the function.
- D. This might be a calculation error (e.g., 24*6 = 144, 144-72 = 72).
Quadratic Function Maximum
For a quadratic function f(x) = ax^2 + bx + c, if 'a' is negative, the parabola opens downwards, and its vertex represents the maximum value of the function.
- Occurs at the vertex, where x = -b/(2a).
- The maximum value is f(-b/(2a)).
- Common in profit maximization or projectile motion problems.
Memory trick: Profit Peak: Vertex is where the money's made!