(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.