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 answer these questions, we will substitute the given values of x into the function C(x) = 2000 + 2x - 0.0001x^2 and calculate the respective costs or marginal costs.
(a) To find the cost for the 1071st disc, substitute x = 1071 into the function C(x):
C(1071) = 2000 + 2(1071) - 0.0001(1071)^2
Calculating the expression gives:
C(1071) = 2000 + 2142 - 0.0001(1071)^2
C(1071) = 4142 - 0.0001(1071)^2
Round to the nearest cent:
C(1071) = $4142
Therefore, the actual cost incurred in producing the 1071st disc is $4142.
Similarly, for the 1891st disc:
C(1891) = 2000 + 2(1891) - 0.0001(1891)^2
Calculating the expression gives:
C(1891) = 2000 + 3782 - 0.0001(1891)^2
C(1891) = 5782 - 0.0001(1891)^2
Round to the nearest cent:
C(1891) = $5782
Therefore, the actual cost incurred in producing the 1891st disc is $5782.
(b) To find the marginal cost when x = 1070, we need to calculate the difference in cost between producing 1071 and 1070 discs. Substitute x = 1070 into the function C(x):
Marginal cost when x = 1070:
MC(1070) = C(1071) - C(1070)
Using the previously calculated values:
MC(1070) = $4142 - C(1070)
Repeat the same process to find the marginal cost when x = 1890:
Marginal cost when x = 1890:
MC(1890) = C(1891) - C(1890)
Using the previously calculated values:
MC(1890) = $5782 - C(1890)
Now, you can calculate the marginal costs when x = 1070 and 1890 by finding the difference between the costs of producing consecutive discs.