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

Answer:

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

Step-by-step 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

$\displaystyle\bar C=\frac{C}{x}$

$\displaystyle\bar C=\frac{0.03x^2-7x+800}{x}$

$\displaystyle\bar C=0.03x-7+\frac{800}{x}$

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

$\displaystyle\bar C'=0.03-\frac{800}{x^2}$

Equating to 0

$\displaystyle 0.03-\frac{800}{x^2}=0$

Solving for x

$\displaystyle x=\sqrt{\frac{800}{0.03}}$

$x=163$

The second derivative is

$\displaystyle\bar C''=\frac{1600}{x^3}$

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

$\displaystyle\bar C(163)=0.03\cdot 163-7+\frac{800}{163}$

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

The minimum average cost is $2.80