To find the minimum unit cost given by the function $C(x) = 0.1x^2 - 28x + 15,318$, we need to find the vertex of the parabola. The formula for the x-coordinate of the vertex of a quadratic function $ax^2 + bx + c$ is given by:

$x = -\frac{b}{2a}$

For the given function, $a = 0.1$ and $b = -28$:

$x = -\frac{-28}{2 \times 0.1} = \frac{28}{0.2} = 140$

Now, substitute $x = 140$ back into the function to find the minimum unit cost:

$C(140) = 0.1(140)^2 - 28(140) + 15,318$

Calculating step by step:

1. $0.1(140)^2 = 0.1 \times 19,600 = 1,960$
2. $28 \times 140 = 3,920$
3. Thus, $C(140) = 1,960 - 3,920 + 15,318 = 13,358$

The minimum unit cost is $13,358.

Would you like more details or have any questions?

Here are five related questions:

1. How does the value of $a$ in the quadratic equation affect the direction of the parabola?
2. What is the significance of the vertex in a quadratic function?
3. How can you determine whether a quadratic function has a minimum or maximum value?
4. How would the minimum unit cost change if the coefficient of $x^2$ were different?
5. What other methods can be used to find the minimum or maximum of a quadratic function?

Tip: The vertex formula $x = -\frac{b}{2a}$ quickly helps find the extremum of a parabola. For a minimum, ensure the parabola opens upward ($a > 0$).