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 the 1891st disc, we need to substitute the values 1071 and 1891 into the function C(x) and round the answers to the nearest cent.
(a)
To find the cost of producing the 1071st disc, substitute x = 1071 into the function C(x):
C(1071) = 2000 + 2(1071) - 0.0001(1071)^2
Calculating this expression, we get:
C(1071) = 2000 + 2142 - 0.0001(1143841)
= 2000 + 2142 - 114.3841
≈ $4127.62
So, the cost incurred in producing the 1071st disc is approximately $4127.62.
To find the cost of producing the 1891st disc, substitute x = 1891 into the function C(x):
C(1891) = 2000 + 2(1891) - 0.0001(1891)^2
Calculating this expression, we get:
C(1891) = 2000 + 3782 - 0.0001(3575881)
= 2000 + 3782 - 357.5881
≈ $9004.41
So, the cost incurred in producing the 1891st disc is approximately $9004.41.
(b)
To find the marginal cost when x = 1070, we need to find the rate of change of the cost function C(x) with respect to x at x = 1070. This can be found by taking the derivative of C(x) with respect to x.
C'(x) = 2 - 0.0002x
Now substitute x = 1070 into the derivative:
C'(1070) = 2 - 0.0002(1070)
= 2 - 0.214
≈ $1.79
So, the marginal cost when x = 1070 is approximately $1.79.
To find the marginal cost when x = 1890, we follow the same steps as above:
C'(x) = 2 - 0.0002x
Substitute x = 1890 into the derivative:
C'(1890) = 2 - 0.0002(1890)
= 2 - 0.378
≈ $1.62
So, the marginal cost when x = 1890 is approximately $1.62.