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=$
(a) To find the actual cost incurred in producing the 1071st and 1891st discs, we need to substitute these values into the cost function C(x) = 2000 + 2x - 0.0001x^2.
For the 1071st disc:
C(1071) = 2000 + 2(1071) - 0.0001(1071^2)
C(1071) = 2000 + 2142 - 123.9391
C(1071) ≈ $3118.06
For the 1891st disc:
C(1891) = 2000 + 2(1891) - 0.0001(1891^2)
C(1891) = 2000 + 3782 - 345.8041
C(1891) ≈ $4436.20
Therefore, the actual cost incurred in producing the 1071st disc is approximately $3118.06, and the actual cost incurred in producing the 1891st disc is approximately $4436.20.
(b) The marginal cost represents the additional cost of producing one additional disc. It can be found by calculating the derivative of the cost function C(x).
First, let's find the derivative of C(x):
C'(x) = 2 - 0.0002x
Now, substitute x = 1070 and x = 1890 into C'(x) to find the marginal cost:
For x = 1070:
C'(1070) = 2 - 0.0002(1070)
C'(1070) ≈ $0.86
For x = 1890:
C'(1890) = 2 - 0.0002(1890)
C'(1890) ≈ $0.62
Therefore, the marginal cost when x = 1070 is approximately $0.86, and the marginal cost when x = 1890 is approximately $0.62.
To find the actual cost incurred in producing the 1071st and 1891st disc, we need to substitute the values of x into the cost function C(x) = 2000 + 2x - 0.0001x^2.
(a) For the 1071st disc:
Substitute x = 1071 into the function:
C(1071) = 2000 + 2(1071) - 0.0001(1071)^2
Calculating this expression will give us the actual cost incurred for the 1071st disc.
(b) For the 1891st disc:
Substitute x = 1891 into the function:
C(1891) = 2000 + 2(1891) - 0.0001(1891)^2
Calculating this expression will give us the actual cost incurred for the 1891st disc.
To find the marginal cost when x = 1070 and 1890, we need to calculate the derivative of the cost function, which represents the rate of change of the cost with respect to the number of discs pressed.
(b) To find the marginal cost at x = 1070:
Take the derivative of the cost function with respect to x:
C'(x) = 2 - 0.0002x
Evaluate this expression at x = 1070 to find the marginal cost at that point.
(b) To find the marginal cost at x = 1890:
Evaluate the derivative at x = 1890 the same way as above to find the marginal cost at that point.
By following these steps, you should be able to find the answers to the questions.