The total weekly cost (in dollars) incurred by Lincoln Records in pressing x compact discs is given by the following function.
C(x) = 2000 + 2x - 0.0001x^2 (0≤x≤6000)
(a) What is the actual cost incurred in producing the 1071st and the 1891st disc? (Round your answers to the nearest cent.)
1071st disc=$
1891st disc=$
(b) What is the marginal cost when x = 1070 and 1890? (Round your answers to the nearest cent.)
1070=$
1890=$
To find the actual cost incurred in producing the 1071st and 1891st discs, we need to substitute the values of x into the cost function C(x).
(a) The cost function is given by: C(x) = 2000 + 2x - 0.0001x^2
Substituting x = 1071 into the cost function:
C(1071) = 2000 + 2(1071) - 0.0001(1071)^2
Calculating this expression will give you the actual cost incurred in producing the 1071st disc.
Substituting x = 1891 into the cost function:
C(1891) = 2000 + 2(1891) - 0.0001(1891)^2
Calculating this expression will give you the actual cost incurred in producing the 1891st disc.
(b) The marginal cost is the rate at which the cost changes with respect to the number of discs produced. To find the marginal cost, we need to find the derivative of the cost function C(x) and then substitute the given values of x.
The derivative of the cost function C(x) is given by: C'(x) = 2 - 0.0002x
Substituting x = 1070 into the derivative:
C'(1070) = 2 - 0.0002(1070)
Calculating this expression will give you the marginal cost when x = 1070.
Substituting x = 1890 into the derivative:
C'(1890) = 2 - 0.0002(1890)
Calculating this expression will give you the marginal cost when x = 1890.
Remember to round your answers to the nearest cent.