To determine the number of airplane engines [tex]\( x \)[/tex] that should be produced to minimize the unit cost, we need to analyze the given cost function:
[tex]\[ C(x) = 0.2x^2 - 120x + 29,420 \][/tex]
### Step-by-Step Solution:
1. Understand the Problem: The given cost function is a quadratic equation, which is a parabolic curve. Since the coefficient of [tex]\( x^2 \)[/tex] is positive ([tex]\( 0.2 > 0 \)[/tex]), the parabola opens upwards, and therefore it has a minimum point.
2. Find the Vertex: For a quadratic function [tex]\( ax^2 + bx + c \)[/tex], the vertex (which gives the minimum or maximum point) can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
In the function [tex]\( C(x) = 0.2x^2 - 120x + 29,420 \)[/tex]:
- [tex]\( a = 0.2 \)[/tex]
- [tex]\( b = -120 \)[/tex]
3. Calculate the Value of x:
[tex]\[ x = -\frac{-120}{2 \times 0.2} = \frac{120}{0.4} = 300 \][/tex]
4. Interpret the Result: The value [tex]\( x = 300 \)[/tex] means that producing 300 airplane engines will minimize the unit cost.
Thus, the number of airplane engines that must be made to minimize the unit cost is:
[tex]\[ \boxed{300} \][/tex]
