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

260 machines for minimum cost.

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.

Answer 2

`x=260\ machines`

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=-(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=-(-156)/(2(0.3))`

`x=260\ machines`

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

`x=260\ machines`