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) = -2x^2 + 20x - 18. What is the maximum profit the company can achieve?
- A32 thousand dollars
- B8 thousand dollars
- C50 thousand dollars
- D10 thousand dollars
Show answer & explanationAnswer & explanation
Correct answer: A. 32 thousand dollars
The profit function is a quadratic equation P(x) = -2x^2 + 20x - 18. Since the leading coefficient (-2) is negative, the parabola opens downward, and the vertex represents the maximum profit. First, find the x-coordinate of the vertex using x = -b/(2a): x = -20/(2*(-2)) = -20/(-4) = 5. This means the maximum profit occurs when 5 units are sold. Now, substitute x=5 back into the profit function to find the maximum profit: P(5) = -2(5)^2 + 20(5) - 18 = -2(25) + 100 - 18 = -50 + 100 - 18 = 50 - 18 = 32.
Why the other options are wrong
- B. Incorrect calculation; perhaps an error in the vertex formula or substitution.
- C. Incorrect calculation; a common error might be miscalculating -2(25) or 20(5).
- D. This is the x-value (number of units) at which the maximum profit occurs, not the maximum profit itself.
Maximum/Minimum of a Quadratic Function (Vertex)
The maximum or minimum value of a quadratic function f(x) = ax^2 + bx + c is the y-coordinate of its vertex, found by evaluating f(-b/(2a)).
- If a > 0, the vertex is a minimum.
- If a < 0, the vertex is a maximum.
- The x-coordinate of the vertex is -b/(2a).
Memory trick: Find the peak's x, then find the peak's y.