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

Explanation:

Step1: Identify the coefficients

The profit - function is a quadratic function $y = ax^{2}+bx + c$, where $a=-18$, $b = 968$, and $c=-7150$.

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 $x=-\frac{b}{2a}$. Substitute $a=-18$ and $b = 968$ into the formula: $x=-\frac{968}{2\times(-18)}=\frac{968}{36}=\frac{242}{9}\approx26.89$.

Step3: Find the maximum profit

Substitute $x = \frac{242}{9}$ into the profit - function $y=-18x^{2}+968x - 7150$: [ \begin{align*} y&=-18\times(\frac{242}{9})^{2}+968\times\frac{242}{9}-7150\ &=-18\times\frac{242^{2}}{81}+\frac{968\times242}{9}-7150\ &=-\frac{2\times242^{2}}{9}+\frac{968\times242}{9}-7150\ &=\frac{-2\times242^{2}+968\times242}{9}-7150\ &=\frac{242(- 2\times242 + 968)}{9}-7150\ &=\frac{242(-484 + 968)}{9}-7150\ &=\frac{242\times484}{9}-7150\ &=\frac{117128}{9}-7150\ &=\frac{117128-64350}{9}\ &=\frac{52778}{9}\ &\approx5864 \end{align*} ]

Answer:

$5864$