9.7: The total cost​ (in dollars) of producing x food processors is ​C(x)=2500+70x−0.1x^2.

Question

The total cost​ (in dollars) of producing x food processors is ​C(x)=2500+70x−0.1x^2.

​(A) Find the exact cost of producing the 71st food processor.

​(B) Use the marginal cost to approximate the cost of producing the 71st food processor.

Answer 1

C=2500+70x - 0.1x^2

C' = 70-0.2x

C'(71) = 70-0.2(71) = 70-14.2 = $55.80 for and estimate of the cost of the 71st food processor

Or you could calculate it exactly by finding C(71) - C(70). Find the total cost of 71

and subtract the total cost of 70

You can see the 2500 will cancel. It's the fixed cost = FC=2500. 70x-0.1x^2 = VC

C(71)-C(70) = VC(71) - VC(70) = 70(71)-0.1(71)^2 - 70(70)+ 0.1(70)^2 =

4970 - 0.1(5041) - 4900 +0.1(4900) = 70-0.01(141) =70-14.1 = $55.90 for cost of 71st processor

There's a dime difference between the estimate and the exact marginal cost

the estimate is the cost at a point. the exact MC is the cost over an interval of one unit