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Katie owns a pretzel stand. Her profit, in dollars, is given by the function , where is the number of pretzels sold. What is the maximum profit , in dollars, Katie can earn?

Sagot :

Using the vertex of the quadratic equation, it is found that the maximum profit she can earn is of $106.

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.

In this problem, her profit is modeled by:

[tex]P(x) = -x^2 + 14x + 57[/tex]

Which is a quadratic equation with coefficients a = -1, b = 14, c = 57, hence, her maximum profit in dollars is given by:

[tex]y_v = -\frac{14^2 - 4(-1)(57)}{4(-1)} = 106[/tex]

More can be learned about the vertex of a quadratic equation at https://brainly.com/question/24737967

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