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=0.04x2−5x+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 $643.75

It should be built 62.5 machines to achieve the minimum average cost

Step-by-step explanation:

The equation that represents the cost C to produce x DVD/BLU-ray players is C = 0.04x² - 5x + 800

To find the minimum cost differentiate C to equate it by 0 to find the average cost per machine and to find the value of the minimum cost

∵ C = 0.04x² - 5x + 800

- Differentiate C with respect to x

∴ $\frac{dC}{dx}=0.04(2)x^{2-1}-5(1)x^{1-1}+0$

∴ $\frac{dC}{dx}=0.08x-5$

- Equate $\frac{dC}{dx}$ by 0

∴ 0.08x - 5 = 0

- Add 5 to both sides

∴ 0.08x = 5

- Divide both sides by 0.08

∴ x = 62.5

That means the minimum average cost is at x = 62.5

Substitute the value of x in C to find the minimum average cost

∵ C = 0.04(62.5)² - 5(62.5) + 800

∴ C = 643.75

∵ C is the average cost

∴ The minimum average cost is $643.75

∵ x is the number of the machines

∴ It should be built 62.5 machines to achieve the minimum

   average cost