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Raquel designs a new hair clip and plans to start a business selling them. She wants to sell her product at a reasonable price...Question
Raquel designs a new hair clip and plans to start a business selling them. She wants to sell her product at a reasonable price while still making the most money possible. She comes up with the quadratic equation y=−0.75x2+24x to reflect her desired revenue. Solving mathematically, if you were to graph the parabola, at what price would the hair clip earn the most revenue?(1 point)$
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GPT-4o mini
To determine the price at which Raquel's hair clip will earn the most revenue, we can analyze the quadratic equation provided, which is:
\[ y = -0.75x^2 + 24x \]
In this equation, \( y \) represents the revenue, and \( x \) represents the price of the hair clip.
The graph of this equation is a parabola that opens downwards (since the coefficient of \( x^2 \) is negative). The vertex of the parabola represents the maximum revenue.
The x-coordinate of the vertex of a quadratic equation in the form \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Here, \( a = -0.75 \) and \( b = 24 \).
Substituting the values into the formula:
\[
x = -\frac{24}{2 \times -0.75}
\]
\[
x = -\frac{24}{-1.5}
\]
\[
x = 16
\]
Thus, the price at which Raquel's hair clip would earn the most revenue is **$16**.
\[ y = -0.75x^2 + 24x \]
In this equation, \( y \) represents the revenue, and \( x \) represents the price of the hair clip.
The graph of this equation is a parabola that opens downwards (since the coefficient of \( x^2 \) is negative). The vertex of the parabola represents the maximum revenue.
The x-coordinate of the vertex of a quadratic equation in the form \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Here, \( a = -0.75 \) and \( b = 24 \).
Substituting the values into the formula:
\[
x = -\frac{24}{2 \times -0.75}
\]
\[
x = -\frac{24}{-1.5}
\]
\[
x = 16
\]
Thus, the price at which Raquel's hair clip would earn the most revenue is **$16**.