The daily total cost (in dollars) incurred by Trappee and Sons for producing x cases of TexaPep hot sauce is given by the following function.
c(x)=0.000002x^3+3x+453
(a) Find the average cost function C.
C = ?
(b) Find the level of production that results in the smallest average production cost. (Round your answer to the nearest whole number.)
? cases
(c) Find the level of production for which the average cost is equal to the marginal cost. (Round your answer to the nearest whole number.)
? cases
To find the average cost function and answer the questions, we need to understand the given information and use some mathematical concepts. Let's go step by step:
(a) First, let's find the average cost function, denoted by C.
The average cost is defined as the total cost divided by the number of cases produced. Mathematically, it can be represented as:
C(x) = c(x) / x
In this case, the total cost function is given as c(x) = 0.000002x^3 + 3x + 453. Therefore, substituting this equation into the average cost function, we have:
C(x) = (0.000002x^3 + 3x + 453) / x
Simplifying this expression, we get:
C(x) = 0.000002x^2 + 3 + 453/x
So, the average cost function is C(x) = 0.000002x^2 + 3 + 453/x.
(b) To find the level of production that results in the smallest average production cost, we need to find the minimum value of the average cost function C(x).
To find the minimum value, we can take the derivative of the average cost function and set it equal to zero. So, let's differentiate C(x) with respect to x:
dC/dx = (d/dx) (0.000002x^2 + 3 + 453/x)
dC/dx = 0.000004x - 453/x^2
Now, set dC/dx equal to zero:
0.000004x - 453/x^2 = 0
Rearrange the equation:
0.000004x = 453/x^2
Multiply both sides of the equation by x^2:
0.000004x^3 = 453
Now, solve for x:
x^3 = 453 / 0.000004
x^3 = 113250000000
Taking the cube root of both sides:
x = (113250000000)^(1/3)
Now, round the answer to the nearest whole number:
x = 494
So, the level of production that results in the smallest average production cost is 494 cases.
(c) To find the level of production for which the average cost is equal to the marginal cost, we need to find the point where the average cost and marginal cost functions are equal.
The marginal cost function is the derivative of the total cost function c(x). In this case, the derivative of c(x) is:
dc(x)/dx = (d/dx) (0.000002x^3 + 3x + 453)
dc(x)/dx = 0.000006x^2 + 3
Now, set the average cost function C(x) equal to the marginal cost function:
0.000002x^2 + 3 + 453/x = 0.000006x^2 + 3
Simplifying the equation, we get:
0.000004x^2 = 453/x
Multiply both sides by x:
0.000004x^3 = 453
x^3 = 453 / 0.000004
x^3 = 113250000000
Taking the cube root of both sides:
x = (113250000000)^(1/3)
Now, round the answer to the nearest whole number:
x = 494
So, the level of production for which the average cost is equal to the marginal cost is 494 cases.