question a company sells widgets. the amount of profit, y, made by the company, is related to the selling…

Explanation:

Step1: Identify the coefficients

The profit - function is a quadratic function $y = ax^{2}+bx + c$, where $a=-33$, $b = 1612$, and $c=-11738$.

Step2: Find the x - value of the vertex

The x - value of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. $x =-\frac{1612}{2\times(-33)}=\frac{1612}{66}=\frac{806}{33}\approx24.42$

Step3: Find the maximum profit

Substitute $x = \frac{806}{33}$ into the profit function $y=-33x^{2}+1612x - 11738$. [ \begin{align*} y&=-33\times(\frac{806}{33})^{2}+1612\times\frac{806}{33}-11738\ &=-33\times\frac{806^{2}}{33^{2}}+\frac{1612\times806}{33}-11738\ &=-\frac{806^{2}}{33}+\frac{1612\times806}{33}-11738\ &=\frac{-806^{2}+1612\times806}{33}-11738\ &=\frac{806\times(- 806 + 1612)}{33}-11738\ &=\frac{806\times806}{33}-11738\ &=\frac{649636}{33}-11738\ &=\frac{649636-11738\times33}{33}\ &=\frac{649636 - 387354}{33}\ &=\frac{262282}{33}\approx7948 \end{align*} ]

Answer:

$7948$