To find the minimum unit cost for manufacturing airplane engines, we need to determine the minimum value of the given quadratic cost function [tex]\( C(x) = 0.7x^2 - 322x + 55046 \)[/tex].
Quadratic functions of the form [tex]\( C(x) = ax^2 + bx + c \)[/tex] open upwards (and therefore have a minimum point) when the coefficient of [tex]\( x^2 \)[/tex] (denoted as [tex]\( a \)[/tex]) is positive. Here, [tex]\( a = 0.7 \)[/tex], which is positive, confirming that the parabola opens upwards.
The vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at the value of [tex]\( x \)[/tex] given by:
[tex]\[
x = \frac{-b}{2a}
\][/tex]
For the function [tex]\( C(x) = 0.7x^2 - 322x + 55046 \)[/tex]:
- [tex]\( a = 0.7 \)[/tex]
- [tex]\( b = -322 \)[/tex]
Substitute these values into the vertex formula to find [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-(-322)}{2 \cdot 0.7} = \frac{322}{1.4} = 230
\][/tex]
So, the number of engines that minimizes the unit cost is [tex]\( x = 230 \)[/tex].
Next, we substitute [tex]\( x = 230 \)[/tex] back into the cost function [tex]\( C(x) \)[/tex] to find the minimum unit cost:
[tex]\[
C(230) = 0.7(230)^2 - 322(230) + 55046
\][/tex]
Let's break this calculation down step-by-step:
1. Calculate [tex]\( (230)^2 \)[/tex]:
[tex]\[
230^2 = 52900
\][/tex]
2. Multiply [tex]\( 0.7 \)[/tex] by [tex]\( 52900 \)[/tex]:
[tex]\[
0.7 \times 52900 = 37030
\][/tex]
3. Multiply [tex]\( 322 \)[/tex] by [tex]\( 230 \)[/tex]:
[tex]\[
322 \times 230 = 74060
\][/tex]
4. Substitute these values into the cost function:
[tex]\[
C(230) = 37030 - 74060 + 55046
\][/tex]
5. Simplify this expression:
[tex]\[
C(230) = 37030 - 74060 + 55046 = 18016
\][/tex]
Thus, the minimum unit cost is [tex]\( C(230) = 18015.999999999993 \)[/tex], which can be approximated to [tex]\( 18016 \)[/tex] without rounding.
Therefore, the minimum unit cost of manufacturing the airplane engines is [tex]\( \$18015.999999999993 \)[/tex] when 230 engines are made.
