The marginal function is define by C’(x) = 6x where C’(x) is the number of hundreds of dollars total cost of “x” hundred of unit of a certain commodity if the cost of 200 unit 2,000 dollars. Find (a) the total cost function (b) the overhead cost.
C(x) = 3x^2 + a
since C(2) = 2000, a = 800
C(x) = 3x^2 + 800
(b) the overhead is C(0), right?
To find the total cost function and the overhead cost, we can follow these steps:
(a) Total Cost Function:
The marginal function C'(x) represents the rate at which the total cost is changing with respect to the number of units produced. To find the total cost function, we need to integrate the marginal function C'(x).
Given: C'(x) = 6x
To find the total cost function, integrate C'(x) with respect to x:
∫C'(x) dx = ∫6x dx
Using the power rule of integration, we have:
C(x) = 3x^2 + C1
Here, C1 is the constant of integration. To determine this constant, we need additional information. In the given problem, it states that the cost of 200 units is $2,000. So we can substitute this information into our total cost function equation to find C1.
C(200) = 3(200)^2 + C1
2,000 = 3(40,000) + C1
2,000 = 120,000 + C1
C1 = -118,000
Therefore, the total cost function (C(x)) is:
C(x) = 3x^2 - 118,000
(b) Overhead Cost:
The overhead cost is a term in the total cost function that remains constant regardless of the number of units produced. In our case, the overhead cost is the value of C1, which we found earlier.
Therefore, the overhead cost is $118,000.
To find the total cost function, we integrate the marginal cost function:
(a) Total Cost Function:
Given that C'(x) = 6x, we can integrate this function to find the total cost function C(x):
∫ C'(x) dx = ∫ 6x dx
C(x) = 3x^2 + C
Since the cost of 200 units is $2,000, we can substitute this value into the equation to find C:
C(200) = 3(200)^2 + C
2000 = 3(40000) + C
2000 = 120000 + C
C = -118000
Therefore, the total cost function C(x) = 3x^2 - 118,000.
(b) Overhead Cost:
To find the overhead cost, we need to subtract the variable cost from the total cost. The variable cost is the cost that varies with the level of production, which in this case is 6x.
Overhead Cost = Total Cost - Variable Cost
= C(x) - C'(x)
= (3x^2 - 118,000) - (6x)
= 3x^2 - 6x - 118,000.