Question

A company estimates that the marginal cost (in dollars per item) of producing x items is 1.73 − 0.004x. If the cost of producing one item is $554, find the cost (in dollars) of producing 100 items. (Round your answer to two decimal places.)

Answer 1

To find the cost of producing 100 items, we can use the marginal cost function provided and integrate it to get the total cost function.

Given the marginal cost function: MC(x) = 1.73 - 0.004x

To find the total cost function (TC(x)), we integrate the marginal cost function with respect to x:

TC(x) = ∫(1.73 - 0.004x) dx

TC(x) = 1.73x - 0.004 * (x^2) + C

where C is the constant of integration.

Now, we can use the given information that the cost of producing one item is $554 to find the value of C:

When x = 1, TC(1) = 554

TC(1) = 1.73 * 1 - 0.004 * (1^2) + C
554 = 1.73 - 0.004 + C
C = 554 - 1.726 = 552.274

Now that we have the value of C, we can find the total cost of producing 100 items (x = 100):

TC(100) = 1.73 * 100 - 0.004 * (100^2) + 552.274

TC(100) = 173 - 0.004 * 10000 + 552.274

TC(100) = 173 - 40 + 552.274

TC(100) = 685.274

The cost of producing 100 items is $685.27 (rounded to two decimal places).