GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A small business sells custom-printed t-shirts. The profit (P) in dollars for selling 'x' shirts can be modeled by the function P(x) = -0.5x² + 50x - 300. How many shirts must be sold to achieve the maximum profit?
- A50 shirts
- B100 shirts
- C25 shirts
- D75 shirts
Show answer & explanationAnswer & explanation
Correct answer: A. 50 shirts
The profit function is a downward-opening parabola (because a = -0.5 is negative). The maximum profit occurs at the x-coordinate of the vertex. The formula for the x-coordinate of the vertex is -b/(2a). Here, a = -0.5 and b = 50. So, x = -50 / (2 * -0.5) = -50 / -1 = 50. Therefore, 50 shirts must be sold for maximum profit.
Why the other options are wrong
- B. This could be an incorrect interpretation or calculation, possibly doubling the correct value.
- C. This could result from an error in the vertex formula calculation, perhaps using 2a as 2*0.5=1 instead of 2*-0.5=-1.
- D. This might be a result of miscalculating the denominator or other algebraic errors.
Quadratic Max/Min Application
For real-world problems modeled by quadratic functions, the maximum or minimum value often represents an optimal condition (e.g., maximum profit, minimum cost, maximum height). This value is found at the vertex of the parabola.
- If the quadratic coefficient 'a' is negative, the vertex gives a maximum value.
- If 'a' is positive, the vertex gives a minimum value.
- The x-coordinate of the vertex, -b/(2a), gives the input value (e.g., number of units) for the max/min.
Memory trick: Find the peak of the profit hill or the bottom of the cost valley.