Using the vertex of a quadratic function, it is found that:
a) The revenue is maximized with 336 units.
b) The maximum revenue is of $56,448.
What is the vertex of a quadratic equation?
A quadratic equation is modeled by:
[tex]y = ax^2 + bx + c[/tex]
The vertex is given by:
[tex](x_v, y_v)[/tex]
In which:
- [tex]x_v = -\frac{b}{2a}[/tex]
- [tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]
Considering the coefficient a, we have that:
- If a < 0, the vertex is a maximum point.
- If a > 0, the vertex is a minimum point.
The demand function is given by:
p(x) = 336 - 0.5x.
Hence, the revenue function is:
R(x) = xp(x)
R(x) = -0.5x² + 336x.
Which has coefficients a = -0.5, b = 336.
Hence, the value of x that maximizes the revenue, and the maximum revenue, are given, respectively, as follows:
- [tex]x_v = -\frac{336}{2(-0.5)} = 336[/tex]
- [tex]y_v = -\frac{336^2 - 4(-0.5)(0)}{4(-0.5)} = 56448[/tex]
More can be learned about the vertex of a quadratic function at https://brainly.com/question/24737967
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