A manufacturer has found that marginal cost is dc/dx=3x^2-60x+400 dollars per unit when x units have been produced. The total cost of producing the first 2 units is $900. What is the total cost of producing the first 5 units?

I would integrate, you are given at x=2 the cost, which lets you determine the constant of integration.

Explain Bot Explain Bot answered
8 months ago

To find the total cost of producing the first 5 units, we can use the given marginal cost function and the information about the cost of producing the first 2 units.

The marginal cost function is given as dc/dx = 3x^2 - 60x + 400 dollars per unit. To find the total cost function, we need to integrate this function with respect to x.

∫(3x^2 - 60x + 400) dx = ∫3x^2 dx - ∫60x dx + ∫400 dx

Integrating, we get:

x^3 - 30x^2 + 400x + C

We know that the total cost of producing the first 2 units is $900. This means that when x = 2, the total cost function equals 900:

2^3 - 30(2)^2 + 400(2) + C = 900

8 - 120 + 800 + C = 900

-112 + C = 900

C = 1012

So the total cost function is:

C(x) = x^3 - 30x^2 + 400x + 1012

To find the total cost of producing the first 5 units, we substitute x = 5 into the total cost function:

C(5) = 5^3 - 30(5)^2 + 400(5) + 1012

C(5) = 125 - 30(25) + 2000 + 1012

C(5) = 125 - 750 + 2000 + 1012

C(5) = 2387

Therefore, the total cost of producing the first 5 units is $2387.

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