Question

a company that produces cell phones has a cost function of $c = 13x^2 - 445x + 22659$, where $c$ is the cost in dollars and $x$ is the number of cell phones produced (in thousands). how many units of cell phone (in thousands) minimizes the cost function?
$x = \square$ thousand phones produced will minimize $c$
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Explanation:

Step1: Identify the function type

The cost function \( C = 13x^2 - 445x + 22659 \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a = 13 \), \( b = - 445 \), and \( c = 22659 \). Since \( a>0 \), the parabola opens upwards, and the vertex of the parabola gives the minimum value of the function.

Step2: Use the vertex formula for x - coordinate

For a quadratic function \( y=ax^{2}+bx + c \), the x - coordinate of the vertex is given by the formula \( x=-\frac{b}{2a} \).

Substitute \( a = 13 \) and \( b=-445 \) into the formula:
\( x =-\frac{-445}{2\times13}=\frac{445}{26}\approx17.115 \)

Answer:

\( \frac{445}{26}\) (or approximately \( 17.12 \))