A company estimates that the marginal cost (in dollars per item) of producing x items is 1.85 − 0.004x. If the cost of producing one item is $550, find the cost of producing 100 items.

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The marginal cost is the gradient of the total cost C at a given number of items produced x.

Write `(dC)/(dx) = M(x)`

Therefore the total cost of producing r items is

`C = int_0^r M(x) dx + I `

where I is the inital cost

We have

`int_0^1 M(x) dx + I = int_0^1 (1.85 -0.004x) dx + I = 550`

Now `int_0^1 (1.85 -0.004x)dx = (1.85x -0.002x^2)|_0^1 = 1.85 - 0.002 = 1.848`

Therefore I = 550 - 1.848 = 548.152

The total cost of producing x = 100 items is given by

`int_0^100 (1.85 -0.004x)dx + 548.152 = (1.85x -0.002x^2)|_0^100 + 548.152`

`= 1.85(100) -0.002(10000) + 548.152 = 713.152`

The total cost of producing 100 items is $713.15

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Marginal cost is defined as the change in cost of production C(x) if the number of items x being produced is changed, MC = C'(x)

In the problem MC = C'(x) = 1.85 - 0.004x

C(x) = 1.85x - 0.002x^2 + C

It is given that the cost of production of 1 unit is $550, this gives C(x) = 1.85x - 0.002x^2 + 550

The cost of producing 100 units is C(100) = $715

The cost of producing 100 items is $715.

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