A company estimates that the marginal cost (in dollars per item) of producing x items is 1.71 − 0.002x (Calculus)

Question

A company estimates that the marginal cost (in dollars per item) of producing x items is 1.71 − 0.002x. If the cost of producing one item is $572, find the cost of producing 100 items. (Round your answer to two decimal places.)

Answer 1

Assume a constant marginal cost, such that the marginal cost of producing 1 item is equal to the marginal cost of producing the 100th item.

 

We know the total cost of producing 1 item ($572) and the marginal cost of producing that additional 1 item (1.71 - 0.002x) so we can take the integral of the marginal cost function to get the total cost function, substitute in the known value of the total cost function at x=1 ($572) to solve for C, and then substitute 100 into this total cost function to find the cost of producing 100 items.

 

1) Solve for total cost function: ∫ 1.71 - 0.002x dx = 1.71x - 0.001x² + C

2) Set total cost function equal to $572 and let x=1 to find C:

    572 = 1.71x - 0.001x² + C --> 572 = 1.71(1) - 0.001(1²) + C --> 572 = 1.71 - 0.001 + C --> C = 570.291

    --> Total Cost = 1.71x - 0.001x² + 570.291

3) Substitute 100 into total cost function to find the total cost of producing 100 items:

    ---> Cost of producing 100 items = 1.71(100) - 0.001(100²) + 570.291 = 171 - 10 + 570.291 = $731.29

 

Intuitively this makes sense, because the cost of producing the first item is $572, so the cost of producing 100 items (or 99 more items) is roughly the same as taking the marginal cost of 1 item (1.71 - 0.002(1) = 1.708) times 99 (169.092) and adding it to the cost of producing the first item (169.092 + 572 = $741.092), which is pretty close to the answer using integrals.

Answer 2

If one item cost $572 then 100 items would cost $57,200.

Or I am reading the problem incorrectly?