Question

A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function c(x) = 0.3x^2 - 156x + 26,657 . How many machines must be made to minimize the unit cost?
Do not round your answer.

Answer 1

Answer:

$x=260\ machines$

Step-by-step explanation:

Note that we have a cudratic function of negative principal coefficient.

The minimum value reached by this function is found in its vertex.

For a quadratic function of the form

$ax ^ 2 + bx + c$

the x coordinate of the vertex is given by the following expression

$x=-\frac{b}{2a}$

In this case the function is:

$C(x) = 0.3x^2 - 156x + 26,657$

So:

$a=0.3\\b=-156\\c=26,657$

Then the x coordinate of the vertex is:

$x=-\frac{-156}{2(0.3)}$

$x=260\ machines$

Then the number of machines that must be made to minimize the cost is:

$x=260\ machines$

Answer 2

Answer:

260 machines for minimum cost.

Step-by-step explanation:

c(x) = 0.3x^2 - 156x + 26.657

Finding the derivative:

c'(x) = 0.6x - 156

0.6x - 156 = 0   for  maxm/minm cost.

x = 156 / 0.6

= 260

The second derivative  is positive (0.6) so this is a minimum.