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 951st and the 2181st disc? (Round your answers to the nearest cent.)
(b) What is the marginal cost when x = 950 and 2180? (Round your answers to the nearest cent.)
(a) do you mean average cost, not actual cost? If so, then for the 951st disc that is just
C(951)/951 = $4.01
(b) dC/dx = 2 + 0.0002x
so for the 951st disc, that would be $2.19
Hmmm. For the actual cost of the nth disc, that would be
C(n) - C(n-1)
so for the 951st disc, that is
C(951)-C(950) = 3811.56 - 3809.75 = $1.81
To find the cost incurred in producing a specific number of compact discs and the marginal cost at a given number of discs, we need to evaluate the function C(x) and its derivative.
(a) To find the cost incurred in producing the 951st and 2181st disc, we can substitute these values into the function C(x) and evaluate.
1. For the 951st disc:
C(951) = 2000 + 2(951) - 0.0001(951)^2
Let's calculate this using a calculator:
C(951) = 2000 + 2(951) - 0.0001(951)^2 = 2000 + 1902 - 0.0001(903601) ≈ $3670.81
The cost incurred in producing the 951st disc is approximately $3670.81.
2. For the 2181st disc:
C(2181) = 2000 + 2(2181) - 0.0001(2181)^2
Again, let's calculate this:
C(2181) = 2000 + 2(2181) - 0.0001(2181)^2 = 2000 + 4362 - 0.0001(4752361) ≈ $8444.54
The cost incurred in producing the 2181st disc is approximately $8444.54.
(b) To find the marginal cost at x = 950 and x = 2180, we need to calculate the derivative of the cost function C(x) with respect to x.
The derivative of C(x) is given by dC(x)/dx = 2 - 0.0002x.
1. When x = 950:
dC(950)/dx = 2 - 0.0002(950)
dC(950)/dx = 2 - 0.19 = 1.81
The marginal cost at x = 950 is approximately $1.81.
2. When x = 2180:
dC(2180)/dx = 2 - 0.0002(2180)
dC(2180)/dx = 2 - 0.436 = 1.564
The marginal cost at x = 2180 is approximately $1.56.
So, the answers to the questions are:
(a) The actual cost incurred in producing the 951st disc is approximately $3670.81, and the cost incurred in producing the 2181st disc is approximately $8444.54.
(b) The marginal cost when x = 950 is approximately $1.81, and the marginal cost when x = 2180 is approximately $1.56.