Answer
a) $\$420$
b) The vertex: $(15,410)$
c) About $95$
Work Step by Step
Let's note the variable by $x$.
Given \begin{equation}
C(x)=\frac{2}{5} x^2-12 x+500,\\
a= 0.4, b= -12, c= 500.
\end{equation} a) Set $x= 20$ to find the cost of producing $20$ backpacks.
\begin{equation}
\begin{aligned}
C(20) & =\frac{2}{5} \cdot 20^2-12 \cdot 20+500 \\
& =420.
\end{aligned}
\end{equation} b) To find the vertex of the cost function, we use $a= 0.4$ and $b= -12$ into the following formula.
$$
\begin{aligned}
& x=\frac{-b}{2 a}=\frac{-(-12)}{2(0.4)}=15 \\
& y=0.4 \cdot 15^2-12\cdot 15+500=410.
\end{aligned}
$$ The vertex is $(15,410)$. The means that the minimum cost of producing $15$ backs is about $\$410$.
c) Set $C(x)= 3000$ to find the values of $x$. $$
\begin{aligned}
& \frac{0.4 x^2-12 x+500}{0.4}=\frac{3000}{0.4} \\
& x^2-30 x+1250=7500\\
& x^2-30 x+1250-7500=0\\
& x^2-30 x-6250=0.
\end{aligned}
$$ $$
\begin{aligned}
x& =\frac{-(-30) \pm \sqrt{(-30)^2-4 \cdot (-6250)}}{2} \\
& =\frac{30 \pm \sqrt{25900}}{2}\\
\end{aligned}
$$ $$
\begin{aligned}
x & =\frac{30 - \sqrt{25900}}{2} \\
& \approx-65.47 \\
x & =\frac{30 + \sqrt{25900}}{2} \\
& \approx 95.47
\end{aligned}
$$ This result shows that the school can get about $95$ backpacks with a budget of $\$3000$.
