Step 1 :The unit cost $C$ is given by the function $C(x)=0.4 x^{2}-96 x+16,541$.
Step 2 :The minimum unit cost corresponds to the minimum point of the function $C(x)$.
Step 3 :This is a quadratic function, and the minimum point of a quadratic function $ax^2 + bx + c$ is given by $-\frac{b}{2a}$.
Step 4 :In this case, $a = 0.4$ and $b = -96$. So, we need to calculate $-\frac{b}{2a}$.
Step 5 :Substitute $a = 0.4$ and $b = -96$ into the formula $-\frac{b}{2a}$, we get $x = 120.0$.
Step 6 :Substitute $x = 120.0$ into the function $C(x)$, we get $C = 10781.0000000000$.
Step 7 :\(\boxed{10781}\) is the minimum unit cost.