Answer to Question #274853 in Calculus for sumu

Question

A small factory producing a single product has weekly fixed costs of production of $2,112 and weekly variable costs of $52x + 3/4 x2, where x is the quantity produced. the capacity of the factory is about 600 units.

Past experience suggests that the product’s price and quantity are linked by the following demand equation: p = 200 - 1/4 x (p, x > 0) where p = $ price/unit and x = quantity sold. You are required to:

(a) Find the level of production at which revenue is maximized

(b) Find any break-even points

Answer 1

(a) The level of production at which revenue is maximized is:

$MR = TR'(x) = (p×x)' = 200 - 0.5x = 0,$

0.5x = 200,

x = 400 units.

(b) The break-even points are:

$x = FC/(p - AVC) = 2,112/(200 - 1/4x - 52 - 3/4x) =2,112/(148 - x),$

$x^2 - 148x + 2,112 = 0,$

$x = (148 \pm 116)/2,$

$x_1 = 132$ units, $x_2 = 16$ units.