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Homework answers / question archive / The cost ,in dollars, of producing x belts is given by C(x)= 683+16x-0
The cost ,in dollars, of producing x belts is given by C(x)= 683+16x-0.077x². Find the rate at which average cost is charging when belts have been produced.
When 256 belts have been produced, the average cost is changing at ---- for each additional belt.
(Round to four decimal places as needed. Do not include the $ symbol in your answer.)
Before we can find how the average cost is changing, we first need to define the average cost function. This can be found by dividing the given cost function by x.
[Math Processing Error]C¯(x)=C(x)x=683+16x−0.077x2x=683x+16−0.077x
The rate of change of a function is defined as the derivative. Thus, we need to differentiate this function in order to find how the average cost is changing. Since the function we have contains only terms with x to a numerical power -- remember that [Math Processing Error]1x is the same as [Math Processing Error]x−1 -- we can use the power rule to do this.
[Math Processing Error]C¯′(x)=−683x2−0.077
Evaluating this at a point will tell us how the average cost is changing at that level of production.
[Math Processing Error]C¯′(256)=−683(256)2−0.077≈−0.08742
Thus, the average cost per belt is decreasing at a rate of 9 cents per belt when 256 have been produced.
