The total cost (in dollars) of manufacturing xx auto body frames is C(x)=40,000+700xC(x)=40,000+700x (A) Find the average cost per unit if 400400 frames are produced

(A) To find the average cost per unit if 400 frames are produced, we first need to find the average cost function, ¯C(x).C¯(x). For C(x)=40,000+700xC(x)=40,000+700x we have¯C(x)=C(x)x¯C(x)=40,000+700xx¯C(x)=40,000x+700C¯(x)=C(x)xC¯(x)=40,000+700xxC¯(x)=40,000x+700

So the average cost per unit if 400 frames are produced is given by¯C(400)=40,000400+700¯C(400)=100+700¯C(400)=800C¯(400)=40,000400+700C¯(400)=100+700C¯(400)=800

The average cost per unit if 400 frames are produced is $800.

(B) The average value of the cost function over the interval (0,400)(0,400) is given by the definite integralCave=1400−0∫4000C(x)dxCave=1400∫4000(40,000+700x)dxCave=1400(40,000x+350x2)???4000Cave=1400(40,000(400)+350(400)2−40,000(0)−350(0)2)Cave=1400(16,000,000+56,000,000)Cave=1400(72,000,000)Cave=180,000Cave=1400−0∫0400C(x)dxCave=1400∫0400(40,000+700x)dxCave=1400(40,000x+350x2)|0400Cave=1400(40,000(400)+350(400)2−40,000(0)−350(0)2)Cave=1400(16,000,000+56,000,000)Cave=1400(72,000,000)Cave=180,000

The average value of the cost function on the interval (0,400)(0,400) is $180,000.

(C) $800 is the average cost per unit at a production level of 400 units. $180,000 is the average value of the total cost as production increases from 0 to 400 units.