Answer
a) $\$148,500$
b) $423$ pairs of sunglasses or $1577$ pairs of sunglasses
c) $1000$ pairs of sunglasses
Work Step by Step
Given \begin{equation}
R(s)=-1.5 s^2+30 s,\\
a= -1.5, b= 30, c= 0.
\end{equation} a) Set $s= 9$ to find the revenue of selling $9$ hundred pairs of sunglasses. \begin{equation}
\begin{aligned}
R(9) & =-1.5 \cdot9^2+30\cdot 9 \\
& =148.5
\end{aligned}
\end{equation} b) Set $R(s)= 100$ to find the values of $s$, which is the number of sunglasses that must be sold to earn a revenue of $\$10,000$ .
\begin{equation}
\begin{aligned}
-1.5 s^2+30 s & =100 \\
\frac{-1.5 s^2+30 s}{-1.5} & =\frac{100}{-1.5}\\
s^2-20 s&=-\frac{1000}{15}=-\frac{200}{3}\\
\left( s^3-20 s+\frac{200}{3}\right)\cdot 3&=0\cdot 3\\
3s^2-60s+200& = 0
\end{aligned}
\end{equation} Solve the equation: $$\begin{aligned}
c& =\frac{-(-60) \pm \sqrt{(-60)^2-4 \cdot (3)(200)}}{2\cdot 3} \\
& =\frac{60 \pm \sqrt{1200}}{6 }\\
& = 10\pm 5.77.
\end{aligned}
$$ $$
\begin{aligned}
s& =10-5.77 \\
& =4.23 \\
s & =10+5.77 \\
& =15.77.
\end{aligned}
$$ The company must sell about $423$ pairs of sunglasses or about $1577$ pairs of sunglasses to have a revenue of $\$10,000$.
c) The vertex of the revenue function will give us the maximum revenue that we are looking to maximize. Use $a= -1.5$ and $b= 30$ into the following formula.
$$
\begin{aligned}
& S=\frac{-b}{2 a}=\frac{-30}{2(-1.5)}=10 \\
& R_{\text {max }}=R(10) \\
&=-1.5(10)^2+30 \cdot(10) \\
&=150.
\end{aligned}
$$ The vertex is $(10,150)$. This means that they must sell $1000$ pairs of sunglasses to maximize revenue.
