U3.L4.HW: Quadratic Functions: Question 25

Question

Answer both to the nearest whole number and explain them afterwards.

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 - 8x + 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?

The minimum average cost is _________

The number of DVD/Blu-ray players that should be built to acheive the minimum is ___________

Answer 1

In an Algebra course, this is solved by Completing the Square. Start by extracting the factor of 0.04:

C = 0.04(x^2 - 200x + 20000)

Working within the parentheses to complete the square for x^2 - 200x, add (200/2)^2 = 10000. There are several ways to do the arithmetic, but you should wind up with:

C = 0.04(x^2 - 200x + 10000 +10000)

Distribute the extra constant term and rewrite the perfect square:

C = 0.04(x^2 - 200x + 10000) + 0.04(10000)

C = 0.04(x - 100)^2 + 400

The minimum value of C occurs when (x - 100) = 0, which gives x = 100 and C = 400. There is no need to round here, since both values are whole numbers.