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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 $).

An aircraft factory manufactures airplane engines. The unit cost C (the cost in dollars to make each airplane engine) depends on the number of engines made. If x engines are made, then the unit cost is given by the function C(x) = 0.1x^2 - 28x + 15,318. What is the minimum unit cost?

Learn how to find the minimum unit cost for airplane engines using the quadratic function C(x) = 0.1x^2 - 28x + 15,318. This step-by-step guide explains key algebraic concepts such as the vertex formula and how to apply it to find the minimum value in a quadratic equation.

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