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

Answer:

$3074$

Explanation:

Step1: Identify the coefficients

The profit - function is $y = - 35x^{2}+1038x - 4878$, where $a=-35$, $b = 1038$, $c=-4878$.

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 $a=-35$ and $b = 1038$ into the formula: $x=-\frac{1038}{2\times(-35)}=\frac{1038}{70}=\frac{519}{35}\approx14.83$.

Step3: Find the maximum profit

Substitute $x = \frac{519}{35}$ into the profit - function $y=-35x^{2}+1038x - 4878$: [ \begin{align*} y&=-35\times(\frac{519}{35})^{2}+1038\times\frac{519}{35}-4878\ &=-35\times\frac{519^{2}}{35^{2}}+\frac{1038\times519}{35}-4878\ &=-\frac{519^{2}}{35}+\frac{1038\times519}{35}-4878\ &=\frac{-519^{2}+1038\times519}{35}-4878\ &=\frac{519\times(-519 + 1038)}{35}-4878\ &=\frac{519\times519}{35}-4878\ &=\frac{269361}{35}-4878\ &=\frac{269361-4878\times35}{35}\ &=\frac{269361 - 170730}{35}\ &=\frac{98631}{35}\approx3074 \end{align*} ] So the maximum amount of profit the company can make is $$3074$.