Question

Raquel designs a new hair clip and plants 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 quadratic equation y = -0.75x^2 + 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?

$___

Answer 1

To find the price at which Raquel's hair clip will earn the most revenue, we can analyze the given quadratic equation for revenue:

\[
y = -0.75x^2 + 24x
\]

In this equation:
- \(y\) represents the revenue.
- \(x\) represents the price of the hair clip.

Since this is a quadratic equation in the form \(y = ax^2 + bx + c\), where \(a = -0.75\) and \(b = 24\), we can determine the vertex of the parabola to find the maximum revenue. The x-coordinate of the vertex of a parabola given by the equation \(y = ax^2 + bx + c\) is calculated using the formula:

\[
x = -\frac{b}{2a}
\]

Substituting our values for \(a\) and \(b\):

\[
x = -\frac{24}{2 \cdot -0.75}
\]

Calculating the denominator:

\[
2 \cdot -0.75 = -1.5
\]

Now, substituting back into the formula:

\[
x = -\frac{24}{-1.5} = \frac{24}{1.5}
\]

Calculating \( \frac{24}{1.5} \):

\[
\frac{24}{1.5} = 16
\]

Thus, the price at which the hair clip would earn the most revenue is:

\[
\boxed{16}
\]