Digital SATMath: Advanced MathMedium
A company's monthly profit, P(x), in thousands of dollars, from selling x units of a product is modeled by the function P(x) = -0.5x^2 + 10x - 30. How many units must be sold to achieve the maximum profit?
- A10 units
- B50 units
- C20 units
- D30 units
Show answer & explanationAnswer & explanation
Correct answer: A. 10 units
The profit function P(x) = -0.5x^2 + 10x - 30 is a quadratic equation, representing a parabola that opens downwards (since the leading coefficient -0.5 is negative). The maximum profit occurs at the vertex of this parabola. The x-coordinate of the vertex is given by the formula x = -b/(2a). Here, a = -0.5 and b = 10. So, x = -10 / (2 * -0.5) = -10 / -1 = 10 units.
Why the other options are wrong
- B. This is the maximum profit value (P(10) = -0.5(10)^2 + 10(10) - 30 = -50 + 100 - 30 = 20), not the number of units.
- C. This could be a distracter from an arithmetic error, such as 10/0.5.
- D. This is a distracter, possibly related to the constant term or another incorrect calculation.
Maximum of a Quadratic Function (x-value)
For a quadratic function f(x) = ax^2 + bx + c with a < 0, the x-coordinate of the vertex, given by -b/(2a), represents the input value at which the function reaches its maximum output.
- Occurs at the vertex of the parabola.
- If 'a' is negative, the parabola opens downward, indicating a maximum.
- The formula -b/(2a) finds the specific input (e.g., units, time) for the maximum.
Memory trick: X marks the spot for the 'top' of the profit hill!