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^2-144x+22,433. What is the minimum unit cost?

Answer 1

Answer:

The company should made 240 copy machines for the minimum unit cost

Explanation:

We have to found what is the minimum quantity of copy machines that should have been made ("x") when the derived of the cost respect to "x" is zero. That calls the First Derivative Test.

It means:

$C(x) = 0.3x^{2} -144x+22433$

So, the derived of C(x) respect to "x" is:

$C'(x)=2(0.3)x-144=0.6x-144$

Now, we make the previous expression equal to zero and clear "x":

$0.6x-144=0\\x=144/0.6\\x=240$

So, now, we know that we will have a maximum or a minimum reached when the company makes 240 copy machines. So, with the Second Derivative Test we are going to find if the cost is maximum or minimum.

Let's find the second derived of the function:

$C(x) = 0.3x^{2} -144x+22433\\C'(x)= 0.6x-144\\C''(x)=0.6$

As the second derived is bigger than zero, the Second Derivative Test says to us that there is a minimum when "x" is equal to 240 units