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?

Answer 1

To find the price at which Raquel's hair clip will earn the most revenue, we need to identify the vertex of the quadratic equation given by:

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

In a quadratic equation of the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:

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

In Raquel's equation, \( a = -0.75 \) and \( b = 24 \). We can substitute these values into the formula:

\[
x = -\frac{24}{2 \times -0.75} = -\frac{24}{-1.5} = 16
\]

Thus, the price at which Raquel will earn the most revenue is at \( x = 16 \).

To verify that this indeed gives the maximum revenue, we can compute the revenue at this price by substituting \( x = 16 \) back into the revenue equation:

\[
y = -0.75(16)^2 + 24(16)
\]
\[
y = -0.75(256) + 384
\]
\[
y = -192 + 384
\]
\[
y = 192
\]

Hence, the maximum revenue occurs when the price of the hair clip is set at **$16**, yielding a maximum revenue of **$192**.