Marginal Cost (Integration)

Q:  Suppose the marginal cost is:  MC = 12 + 8x with C(10) = 1020.  Find the fixed and total cost of producing 80 units.

Answer:

We gotta know that the marginal cost is derivative of the cost.  So, to find the cost function, we need to integrate the marginal cost function:

int[12 + 8x] = 12x + 4x2 + C

So, the cost function, C(x) = 12x + 4x2 + C

And we know that C(10) = 1020.  So, we can plug in these values to solve for the constant C like so:

1020 = 12(10) + 4(10)2 + C

1020 = 120 + 400 + C

1020 = 520 + C

500 = C

So, the cost equation is:

C(x) = 12x + 4x2 + 500

The “C” value represents the “fixed costs”.  So, the fixed costs are $500.

The total cost of producing 80 units can be found by calculating C(80):

C(80) = 12(80)+4(80)^2+500 = $27060

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