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Difficulty: MediumQuadratic Equations

A retail store models its weekly profit, P(d)P(d), in dollars, from selling a certain product at a discount of dd dollars using the function P(d)=−5(d−8)2+2,500P(d) = -5(d - 8)^2 + 2,500, where 0≤d≤200 \leq d \leq 20. Which of the following is the best interpretation of the value 8 in this context?

  1. A
    The maximum weekly profit, in dollars, that the store can achieve.
  2. The discount, in dollars, that results in the maximum weekly profit.Answer
  3. C
    The weekly profit, in dollars, when no discount is offered.
  4. D
    The discount, in dollars, that results in the minimum weekly profit.

Answer

The discount, in dollars, that results in the maximum weekly profit.
The quadratic function P(d)=−5(d−8)2+2,500P(d) = -5(d - 8)^2 + 2,500 is given in vertex form, P(d)=a(d−h)2+kP(d) = a(d - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In this equation, the vertex is (8,2500)(8, 2500). The leading coefficient is −5-5, which is negative, meaning the parabola opens downward and the vertex represents the maximum point of the function. In this context, dd is the discount in dollars and P(d)P(d) is the weekly profit. Therefore, the value 8 represents the discount of 8 dollars that results in the maximum weekly profit.

Step-by-Step Solution

1
Identify the form of the quadratic function.
The function P(d)=−5(d−8)2+2,500P(d) = -5(d - 8)^2 + 2,500 is in vertex form, P(d)=a(d−h)2+kP(d) = a(d - h)^2 + k, where (h,k)(h, k) represents the vertex of the parabola.
Recognizing the vertex form allows us to directly identify the vertex coordinates without expanding the equation.
2
Determine the vertex and the direction the parabola opens.
The vertex is (8,2,500)(8, 2,500). Since the leading coefficient a=−5a = -5 is negative, the parabola opens downward, meaning the vertex represents a maximum point.
The sign of the leading coefficient determines whether the vertex represents a maximum or a minimum.
3
Interpret the coordinates of the vertex in context.
The variable dd is the discount in dollars, and P(d)P(d) is the weekly profit in dollars. Thus, at the vertex, the discount is 8 dollars, which corresponds to the maximum profit of 2,500 dollars.
Matching the coordinates of the vertex to the variables in the context gives the correct real-world meaning.

Key Concept

Interpreting the vertex of a quadratic function in vertex form within a real-world context.
Estimated Time:1m 0s
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Topics

Quadratic Equations

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