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 - 0.0001x2
(0 x 6000)
(a) What is the actual cost incurred in producing the 1011st and the 1821st disc? (Round your answers to the nearest cent.)
1011st disc $ ??
1821st disc $ ??
(b) What is the marginal cost when x = 1010 and 1820? (Round your answers to the nearest cent.)
1010 $ ??
1820 $ ??
To solve this problem, we need to evaluate the given cost function for different values of x.
(a) To find the cost incurred in producing the 1011st and the 1821st disc, we can plug in the values of x into the cost function C(x) = 2000 + 2x - 0.0001x^2.
For the 1011st disc:
C(1011) = 2000 + 2(1011) - 0.0001(1011)^2
= 2000 + 2022 - 0.0001(1022121)
= 2000 + 2022 - 102.2121
= 4044 - 102.2121
= 3941.7879
The cost incurred in producing the 1011st disc is approximately $3941.79.
For the 1821st disc:
C(1821) = 2000 + 2(1821) - 0.0001(1821)^2
= 2000 + 3642 - 0.0001(3315241)
= 2000 + 3642 - 331.5241
= 5642 - 331.5241
= 5310.4759
The cost incurred in producing the 1821st disc is approximately $5310.48.
(b) The marginal cost represents the rate of change of the cost with respect to the number of discs produced. To find the marginal cost when x = 1010 and 1820, we need to calculate the derivative of the cost function C(x).
C(x) = 2000 + 2x - 0.0001x^2
Differentiating C(x) with respect to x gives:
C'(x) = 2 - 0.0002x
Now, we can substitute x = 1010 and x = 1820 into C'(x) to find the marginal cost.
When x = 1010:
C'(1010) = 2 - 0.0002(1010)
= 2 - 0.202
= 1.798
The marginal cost when x = 1010 is approximately $1.80.
When x = 1820:
C'(1820) = 2 - 0.0002(1820)
= 2 - 0.364
= 1.636
The marginal cost when x = 1820 is approximately $1.64.
Therefore, the answers to the questions are:
(a) The cost incurred in producing the 1011st disc is approximately $3941.79, and the cost incurred in producing the 1821st disc is approximately $5310.48.
(b) The marginal cost when x = 1010 is approximately $1.80, and the marginal cost when x = 1820 is approximately $1.64.