To determine the minimum unit cost for manufacturing copy machines given the cost function [tex]\( C(x) = 0.8x^2 - 256x + 40,343 \)[/tex], we can follow these steps:
1. Understand the Cost Function:
The cost function is a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 0.8 \)[/tex], [tex]\( b = -256 \)[/tex], and [tex]\( c = 40,343 \)[/tex].
2. Identify the Vertex of the Parabola:
Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 0.8 \)[/tex]) is positive, the parabola opens upwards. The minimum value of the quadratic function occurs at the vertex of the parabola.
3. Calculate the x-value of the Vertex:
The x-coordinate of the vertex of a parabola given by [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substituting [tex]\( a = 0.8 \)[/tex] and [tex]\( b = -256 \)[/tex],
[tex]\[
x = -\frac{-256}{2 \times 0.8} = \frac{256}{1.6} = 160
\][/tex]
4. Calculate the Minimum Unit Cost:
Now, we substitute [tex]\( x = 160 \)[/tex] back into the original cost function [tex]\( C(x) \)[/tex] to find the minimum unit cost.
[tex]\[
C(160) = 0.8(160)^2 - 256(160) + 40,343
\][/tex]
First, calculate [tex]\( (160)^2 \)[/tex]:
[tex]\[
(160)^2 = 25,600
\][/tex]
Then,
[tex]\[
0.8 \times 25,600 = 20,480
\][/tex]
And,
[tex]\[
256 \times 160 = 40,960
\][/tex]
Therefore,
[tex]\[
C(160) = 20,480 - 40,960 + 40,343
\][/tex]
Simplifying,
[tex]\[
20,480 - 40,960 = -20,480
\][/tex]
Adding 40,343,
[tex]\[
-20,480 + 40,343 = 19,863
\][/tex]
Therefore, the minimum unit cost [tex]\( S \)[/tex] is $19,853.
