A financial analyst is modeling the quarterly profit of a company. The profit P(x) (in thousands of dollars) from selling 'x' hundreds of units is modeled by the function P(x) = -2x^2 + 20x - 18. What is the maximum profit the company can achieve?
- A$32,000
- B$68,000
- C$50,000
- D$18,000
Show answer & explanationAnswer & explanation
Correct answer: A. $32,000
The profit function is a quadratic equation in the form P(x) = ax^2 + bx + c, where a = -2, b = 20, and c = -18. Since 'a' is negative, the parabola opens downwards, indicating that the vertex represents the maximum point. The x-coordinate of the vertex is given by -b/(2a), which will give the number of units for maximum profit. Substituting this x-value into the profit function P(x) will yield the maximum profit.
Why the other options are wrong
- B. This value is too high and likely results from a significant calculation error or a misunderstanding of how to find the maximum of a quadratic function.
- C. This value is plausible but incorrect; it might result from an incorrect calculation of the vertex or a misunderstanding of the formula.
- D. This value might be derived from a calculation error or a misinterpretation of the y-intercept. The y-intercept represents the profit when 0 units are sold, which is -18 (or a loss of $18,000), not the maximum profit.
Maximum/Minimum of a Quadratic Function
For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex represents the maximum or minimum value of the function. If a > 0, the parabola opens upwards and the vertex is a minimum. If a < 0, the parabola opens downwards and the vertex is a maximum.
- The x-coordinate of the vertex is given by the formula x = -b/(2a).
- The y-coordinate of the vertex is found by substituting the x-coordinate back into the function: y = f(-b/(2a)).
- This concept is crucial for optimization problems in various fields like business, engineering, and physics.
Memory trick: Vertex Vibe: 'a' decides the curve, -b/2a finds the peak!