Question

a certain companys main source of income is selling cloth bracelets. the companys annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by: $p(x) = -2x^2 + 16x - 24$ what bracelet price should the company set to earn this maximum profit? dollars

Explanation:

Step1: Identify the quadratic function form

The profit function \( P(x) = -2x^2 + 16x - 24 \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a = -2 \), \( b = 16 \), and \( c = -24 \). For a quadratic function, the vertex (which gives the maximum or minimum) occurs at \( x = -\frac{b}{2a} \). Since \( a < 0 \), the parabola opens downward, so the vertex is the maximum point.

Step2: Calculate the x-coordinate of the vertex

Using the formula \( x = -\frac{b}{2a} \), substitute \( a = -2 \) and \( b = 16 \):

$$ x = -\frac{16}{2 \times (-2)} = -\frac{16}{-4} = 4 $$

Answer:

4