Question

1-Suppose the total cost function for manufacturing a certain product is c(x)=.2(0.01x^2+120) dollars where x represent the number of units produced. Find the level ofproduction that will minimize the average cost. Show your calculus. Round to the nearest integer. You can use either the first derivative test or the second derivativetest to show this is a minimum.

2- The daily total cost (in dollars) incurred by Trappee and Sons for producing x cases of Texa Pep hot sauce is given by the function c(x) = 0.000002x^3+5x+400

a.) Find the average cost function C(x) .
b.) Find the level of production that results in the smallest average production cost.
c.) Find the level of production for which the average cost is equal to the marginal cost.

Answer 1

1. avg cost fn= C(x) = 0.002x + 24/x

to minimize the average cost, A'(x)=0

C'(x)= 0.002-24/x²= 0

∴ x=109.54

C''(x) = 48/x³ > 0

hence it is a minimum

∴x = 110

2.

a) avg cost fn= C(x)= 0.000002x² + 5+400/x

to minimize the average cost, C'(x)=0

C'(x)=0.000004x-400/x²= 0

∴x=464.16

C''(x) = 0.000004 +800/x³> 0

hence it is a minimum

b) ∴x = 464

c) C(x) =marginal cost= C'(x) =

0.000004x-400/x² =0.000002x²+ 5+400/x

0.000002x⁴-0.000004x³+5x²+ 400x -400 =0

solving it ,we get the answer

Answer 2

2nd part :


a) To find the average cost function, you divide the cost function by x.
C(x)/x = (0.000002x^3+5x+400)/x
= 0.000002x^2 + 5 + 400x^-1

b) Find the derivative of Cbar(x). Set it equal to zero. Verify that it is a min.

c) Take the derivative of C(x) (this is known as the marginal cost) and set it equal to Cbar(x)