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

Question

𝑦
=
−
0.75
𝑥
2
+
24
𝑥
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 that maximizes Raquel's revenue represented by the quadratic equation:

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

we need to determine the vertex of this parabola. The general form of a quadratic equation is given by:

\[ y = ax^2 + bx + c \]

where \(a = -0.75\), \(b = 24\), and \(c = 0\). The x-coordinate of the vertex, which gives the maximum value of the function when \(a < 0\), can be found using the formula:

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

Substituting the values of \(a\) and \(b\):

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

Calculating the denominator:

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

Then,

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

So, the price \(x\) that maximizes the revenue \(y\) is \( \boxed{16} \).

To find the maximum revenue, substituting \(x = 16\) back into the revenue equation:

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

Calculating \(16^2\):

\[ 16^2 = 256 \]

Now substitute this in:

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

Thus, the maximum revenue is \(192\) when the price of the hair clip is set at \( \boxed{16} \).