Final answer:
The marginal cost function is C'(x) = 90 - 0.1x. The marginal cost at a producing level of 500 tanks/week is $40. The exact cost of producing the 501st item is $47049.5.
Step-by-step explanation:
To find the marginal cost function, we need to find the derivative of the total cost function with respect to the quantity produced, x.
So, taking the derivative of C(x) = 10000 + 90x - 0.05x^2:
C'(x) = 90 - 0.1x
Hence, the marginal cost function is C'(x) = 90 - 0.1x.
To find the marginal cost at a producing level of 500 tanks/week, we substitute x = 500 into the marginal cost function:
C'(500) = 90 - 0.1(500) = 90 -50 = 40.
Therefore, the marginal cost at a producing level of 500 tanks/week is $40.
To find the exact cost of producing the 501st item, we substitute x = 501 into the total cost function:
C(501) = 10000 + 90(501) - 0.05(501)^2 = 47049.5.
Therefore, the exact cost of producing the 501st item is $47049.5.