How many DVD/Bluray players were produced based on marginal cost?

Question

A certain electronics manufacturer found that the marginal cost  to produce x DVD/Blu- ray players can be found using the equation . If the marginal cost were $618, how many DVD/Blu-ray players were produced?

Please explain each step, thank you. The videos my professor provides are incredibly unhelpful.

Answer 1

We have been given a cost function C = 0.03x² - 7x + 800 which is a quadratic equation.

And we've been asked to find x (quantity of players produced) when C = $618.

Graphically, you can plot two functions: y = 0.03x² - 7x + 800 which is a parabola, and y = 618 which is a horizontal line. Most likely (as there's a parabola) there will be two points of intersection. We want to find where they intersect. You can use Demos Graphing or similar to see what this looks like.

Given the quadratic and the required C value, we can say that 618 = 0.03x² - 7x + 800

=> 0.03x² - 7x + 800 - 618 = 0

=> 0.03x² - 7x + 182 = 0

this fits the quadratic standard form ax² + bx + c where a = 0.03 b = -7 and c = 182 so we can use the quadratic formula (as I don't think it will simplify/factorize) to find x.

x = -b / _2a ± √(b² - 4ac) / _2a

=> x = -(-7) / _2(0.03) ± √( (-7)² - 4(0.03)(182) ) / _2(0.03)

=> x = 7 / _0.06 ± √( 49 - 21.84) / _0.06

=> x = 116.667 ± √( 27.16) / _0.06

=> x = 116.6667 + 86.8588 or x = 116.667 - 86.859

=> x = 203.526 (3 dp) or x = 29.808 (3 dp)

Hence, to nearest unit (as we can't make fractional units), we can produce 30 or 204 units.

Double-check with original equation and found x values:

0.03(30)² - 7(30) + 800 = 617

0.03(204)² - 7(204) + 800 = 620.48

As these unit numbers are quite low, I would double-check whether x is units, or thousands of units. In which case, we have 29,808 units or 203,526 units.

and

0.03(29.808)² - 7(29.808) + 800 = 617.9995

0.03(203.526)² - 7(203.526) + 800 = 618.0029