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.
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.