The marginal cost function of manufucturing x units of a commodity is 6+10x-6x^2.
Find the total cost and average cost. Given that the total cost of producing 1 unit is 15.
To find the total cost of manufacturing x units of a commodity, we need to integrate the marginal cost function.
TC(x) = ∫(6 + 10x - 6x^2) dx
To find the indefinite integral of each term, we add the integral of 6 with respect to x, the integral of 10x with respect to x, and the integral of -6x^2 with respect to x:
TC(x) = 6x + 5x^2 - 2x^3 + C
Where C is the constant of integration. Since we know that the total cost of producing 1 unit is 15, we can substitute this into the equation and solve for C:
15 = 6(1) + 5(1)^2 - 2(1)^3 + C
15 = 6 + 5 - 2 + C
15 = 9 + C
C = 15 - 9
C = 6
Now we can write the total cost function as:
TC(x) = 6x + 5x^2 - 2x^3 + 6
To find the average cost, we divide the total cost (TC(x)) by the number of units (x):
AC(x) = TC(x)/x
AC(x) = (6x + 5x^2 - 2x^3 + 6)/x
AC(x) = 6 + 5x - 2x^2 + 6/x
So, the total cost function is 6x + 5x^2 - 2x^3 + 6, and the average cost function is 6 + 5x - 2x^2 + 6/x.