Answer
a) $\$2975$
b) Vertex: $(50,2875)$
c) No amount of uniforms can be made of a budget of $\$1600$
Work Step by Step
Given \begin{equation}
C(u)=\frac{1}{4} u^2-25 u+3500.
\end{equation} a) Set $u= 30$ to find the cost of producing $30$ uniforms.
$$
\begin{aligned}
C(30) & =\frac{1}{4} \cdot 30^2-25 \cdot 30+3500 \\
& =2975.
\end{aligned}
$$ b) To find the vertex of the cost function, we use $a= 1/4$ and $b= -25$ into the following formula. $$
\begin{aligned}
& x=\frac{-b}{2 a}=\frac{-(-25)}{2(0.25)}=50 \\
& y=\frac{1}{4} \cdot 50^2-25\cdot 50+3500=2875.
\end{aligned}
$$ The vertex is $(50,2875)$. The means that the minimum cost of producing $50$ uniforms is about $\$2875$.
c) Set $C(u)= 1600$ to find the values of $u$.
$$
\begin{aligned}
\frac{1}{4} u^2-25 u+3500&=1600\\
\left(\frac{1}{4} u^2-25 u+3500\right)\cdot 4&=1600\cdot 4 \\
u^2-100 u+14000&=6400 \\
u^2-100 u+14000-6400&= 0\\
u^2-100 u+7600&= 0\\
u& =\frac{-(-100) \pm \sqrt{100^2-4 \cdot 7600}}{2} \\
& =\frac{100 \pm \sqrt{-20400}}{2}.
\end{aligned}
$$ This result shows that it is impossible to produce any number of uniform with a budget of $\$1600$.
