Key Characteristics of Quadratics

Question

A medical equipment industry manufactures X-ray machines. The unit cost C

 (the cost in dollars to make each X-ray machine) depends on the number of machines made. If x

 machines are made, then the unit cost is given by the function =Cx+−0.8x2320x45,930

. How many machines must be made to minimize the unit cost?



Answer 1

C=-.8x^2 + 320x +45,930

constant term= fixed cost

x & x^2 terms =variable cost

x^2 term indicates economies of scale

C'= -1.6x +320=0

x=320/1.6= 200

maximum cost= -.8(200)^2+320(200)+45930 = -32,000+64000+45,930= $77,930

max per unit cost=C/x

= 77930/200=$389.65

or

C/x= -.8x+320+45930/x=-.8(200)+320+45930/200=

-160+320+229.65

=$389.65= max per unit cost

minimum per unit cost is at an infinite ouput level= negative $infinity, due to impossibly large economies of scale

UNLESS you more realistically meant to write

C(x)=.8x^2-320x+45930, with diseconomies of scale when x>200

then

C'=1.6x-320=0

x=200 X-ray machines

min cost= $33,930 less than fixed costs

min per unit cost=33930/200= $169.65 per X-ray machine