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A supply company manufactures copy machines. The unit cost [tex]\(C\)[/tex] (the cost in dollars to make each copy machine) depends on the number of machines made. If [tex]\(x\)[/tex] machines are made, then the unit cost is given by the function [tex]\(C(x) = 0.7x^2 - 378x + 55,995\)[/tex].

How many machines must be made to minimize the unit cost?

Do not round your answer.



Answer :

GinnyAnswer
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.