Question

A certain electronics manufacturer found that the average cost C to produce x DVD/Blu- ray players can be found using the equation C=0.03x2−7x+800. What is the minimum average cost per machine and how many DVD/Blu-ray players should be built in order to acheive that minimum?

Answer 1

The minimum average cost is $2.80 when 163 players are built

Explanation:

Average Cost Function

We'll assume the given function as the total cost to produce x players, and NOT the average cost since that is a different definition, as shown below.

The cost C to produce x DVD/Blu- ray players is given by the equation

`C=0.03x^2-7x+800`

The Average Cost function is defined as

`\displaystyle\bar C=(C)/(x)`

`\displaystyle\bar C=(0.03x^2-7x+800)/(x)`

`\displaystyle\bar C=0.03x-7+(800)/(x)`

To find the extreme value of the average cost, we must take the first derivative of the function

`\displaystyle\bar C'=0.03-(800)/(x^2)`

Equating to 0

`\displaystyle 0.03-(800)/(x^2)=0`

Solving for x

`\displaystyle x=\sqrt{(800)/(0.03)}`

`x=163`

The second derivative is

`\displaystyle\bar C''=(1600)/(x^3)`

For x=163 the second derivative is positive, thus x=163 is a minimum value. Let's compute the minimum average cost

`\displaystyle\bar C(163)=0.03\cdot 163-7+(800)/(163)`

`\displaystyle\bar C(163)=2.80`

The minimum average cost is $2.80