Answered

$\begin{array}{c}

y=-6 x^2+100 x-180 \\

x=\text{selling price of each soccer ball} \\

y=\text{daily profit from soccer balls}

\end{array}$



The vertex is located at $(8.33,236.67)$. What does this point represent in context? Choose two correct answers.


  • The maximum daily profit from the sale of soccer balls is \$8.33.
  • The maximum profit occurs when the store charges \$8.33 per soccer ball.
  • The minimum daily profit from the sale of soccer balls is \$236.67.
  • The vertex is a minimum of the graph.
  • The maximum daily profit from the sale of soccer balls is \$236.67.

Verified Answer :

  • The function \(y = -6x^2 + 100x - 180\) represents a downward-opening parabola because the leading coefficient is negative (\(-6 < 0\)).
  • The vertex of this parabola at \((8.33, 236.67)\) signifies the absolute maximum of the daily profit function.
  • The \(x\)-coordinate of \(8.33\) represents the optimal selling price per soccer ball, while the \(y\)-coordinate of \(236.67\) represents the resulting maximum daily profit.
  • The two correct interpretations are:

1) The maximum profit occurs when the store charges \(\$8.33\) per soccer ball.
2) The maximum daily profit from the sale of soccer balls is \(\$236.67\).

  • Final Answer: \(\boxed{\text{The maximum profit occurs when the store charges \$8.33 per soccer ball, and the maximum daily profit from the sale of soccer balls is \$236.67.}}\)

Explanation

  1. Analyzing the Profit Function and the Vertex

The given profit function is a quadratic equation \(y = -6x^2 + 100x - 180\), where \(x\) represents the selling price of each soccer ball in dollars and \(y\) represents the daily profit in dollars. The vertex of the parabola, given as \((8.33, 236.67)\), is the point where the function reaches its maximum or minimum value. In a quadratic function of the form \(y = ax^2 + bx + c\), if the coefficient \(a\) is negative, the parabola opens downward, which means the vertex is the highest point on the graph, representing the maximum value of the function. Here, \(a = -6\), which is less than 0, confirming that the parabola opens downward. Therefore, the vertex represents the maximum profit.

  1. Interpreting the Vertex Coordinates in Context

Since the vertex \((8.33, 236.67)\) represents the maximum point on the graph, the \(x\)-coordinate (8.33) represents the independent variable at which this maximum occurs, and the \(y\)-coordinate (236.67) represents the dependent variable (the maximum value itself).

  • The \(x\)-coordinate, \(x = 8.33\), corresponds to the selling price of each soccer ball that maximizes the profit. So, charging \(\$8.33\) per soccer ball yields the maximum profit.
  • The \(y\)-coordinate, \(y = 236.67\), corresponds to the daily profit value at that specific price. So, the maximum daily profit from the sale of soccer balls is \(\$236.67\).
  1. Identifying Correct Interpretations

Based on our analysis, we can determine the correct statements from the provided options:

  • The maximum profit occurs when the store charges \(\$8.33\) per soccer ball. (Correct)
  • The maximum daily profit from the sale of soccer balls is \(\$236.67\). (Correct)

These two statements accurately describe the vertex \((8.33, 236.67)\) within the context of the given profit function.

Examples

Understanding quadratic functions and their vertices is essential for optimizing outcomes in various real-life scenarios. For example, a business owner can model their profit as a function of the price they charge for a product. By finding the vertex of the resulting parabolic graph, they can identify the exact price that will maximize their total profit, ensuring that they are neither charging too much (reducing demand) nor too little (reducing margin). This mathematical approach helps in making data-driven decisions for better business performance.

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