To determine the number of machines that need to be manufactured to minimize the unit cost, we need to analyze the given cost function:
[tex]\[ C(x) = 0.7x^2 - 378x + 55995 \][/tex]
This function is a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 0.7 \)[/tex], [tex]\( b = -378 \)[/tex], and [tex]\( c = 55995 \)[/tex]. In a quadratic equation, the minimum or maximum value occurs at the vertex of the parabola.
For a quadratic function, the x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in our values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{-378}{2 \cdot 0.7} \][/tex]
First, simplify the denominator:
[tex]\[ 2 \cdot 0.7 = 1.4 \][/tex]
Then, compute:
[tex]\[ x = \frac{378}{1.4} \][/tex]
To find this value:
[tex]\[ x = \frac{378}{1.4} = 270 \][/tex]
So, the number of machines that must be manufactured to minimize the unit cost is 270.
In summary, to minimize the unit cost, the company should manufacture 270 copy machines.
