the revenue, r(x), from producing and selling x awesome hearing aids is modeled by the function r(x)= - 2x^2…

Explanation:

Step1: Identify the coefficients

The revenue function is $R(x)=-2x^{2}+468x$, where $a = - 2$, $b = 468$ and $c = 0$. For a quadratic - function $y = ax^{2}+bx + c$, the x - coordinate of the vertex (which gives the value of $x$ for maximum or minimum) is $x=-\frac{b}{2a}$.

Step2: Calculate the number of hearing - aids for maximum revenue

Substitute $a=-2$ and $b = 468$ into the formula $x =-\frac{b}{2a}$. $x=-\frac{468}{2\times(-2)}=\frac{468}{4}=117$.

Step3: Calculate the maximum revenue

Substitute $x = 117$ into the revenue function $R(x)=-2x^{2}+468x$. $R(117)=-2\times(117)^{2}+468\times117$. First, calculate $(117)^{2}=13689$. Then $-2\times(117)^{2}=-2\times13689=-27378$. And $468\times117 = 54756$. $R(117)=-27378 + 54756=27378$.

Answer:

The number of hearing aids to be produced and sold to maximize the revenue is 117. The maximum revenue is 27378.