The unit cost function is given by:

$C(x) = 0.8x^2 - 224x + 29,372$

This is a quadratic function, and since the coefficient of $x^2$ is positive (0.8), the parabola opens upwards, meaning that the vertex of the parabola will give us the minimum value of the unit cost.

The formula to find the x-coordinate of the vertex of a quadratic function $ax^2 + bx + c$ is:

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

In this case, $a = 0.8$ and $b = -224$, so we substitute these values into the formula:

$x = \frac{-(-224)}{2(0.8)} = \frac{224}{1.6} = 140$

Thus, 140 engines must be made to minimize the unit cost.

Would you like more details on this calculation or related concepts?

Here are 5 related questions:

1. What is the minimum unit cost after producing 140 engines?
2. How does the unit cost change if fewer or more than 140 engines are produced?
3. How is the vertex formula derived?
4. Can you apply this concept to other manufacturing cost functions?
5. What does the term "marginal cost" mean in production cost analysis?

Tip: The vertex of a quadratic function is a powerful tool for finding minimum or maximum values. Always check the sign of the $x^2$ coefficient to know if it's a minimum (positive) or maximum (negative).