The given equation [tex]\( y = -6x^2 + 100x - 180 \)[/tex] represents the relationship between the selling price of each soccer ball (denoted by [tex]\( x \)[/tex]) and the daily profit (denoted by [tex]\( y \)[/tex]) from selling the soccer balls.
To understand what the vertex [tex]\( (8.33, 236.67) \)[/tex] represents in this context, follow these steps:
1. Identify the Vertex:
The vertex form of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] provides critical information about the maximum or minimum point of the quadratic function. Here, [tex]\( a = -6, b = 100, \)[/tex] and [tex]\( c = -180 \)[/tex]. The vertex of this parabola, given as [tex]\( (8.33, 236.67) \)[/tex], represents the point at which the function reaches its maximum value.
2. Interpret the X-Coordinate of the Vertex:
The [tex]\( x \)[/tex]-coordinate of the vertex, [tex]\( x = 8.33 \)[/tex], represents the selling price of each soccer ball that maximizes the daily profit.
3. Interpret the Y-Coordinate of the Vertex:
The [tex]\( y \)[/tex]-coordinate of the vertex, [tex]\( y = 236.67 \)[/tex], represents the corresponding maximum daily profit when the soccer balls are sold at [tex]\( x = 8.33 \)[/tex] dollars each.
4. Combine These Interpretations:
Therefore, the vertex [tex]\( (8.33, 236.67) \)[/tex] indicates that when the soccer balls are sold at [tex]$8.33 each, the daily profit is maximized, reaching an amount of $[/tex]236.67.
In summary, in the context of the given equation [tex]\( y = -6x^2 + 100x - 180 \)[/tex], the point [tex]\( (8.33, 236.67) \)[/tex] represents the optimal selling price of [tex]$8.33 per soccer ball, which results in the highest possible daily profit of $[/tex]236.67.
