A company produces x units of output at a total cost of 1/3 x^3 – 18x^2 + 160x. Output at which average cost is equal to marginal cost is:

Question

Illustrate an image representing a company's output and cost scenario in an economic context. Incorporate elements like graphs showing a cubic function curve, representing total cost function (1/3 x^3 - 18x^2 + 160x), and intersecting lines to suggest the point of equilibrium where average cost equals marginal cost. Make sure to depict not only the financial elements but also the industrial environment with machinery and workers. Also, include symbolic elements like scales representing balancing cost. Ensure that the image contains no text.

A company produces x units of output at a total cost of 1/3 x^3 – 18x^2 + 160x. Output at which average cost is equal to marginal cost is:

Answer 1

To find the output at which average cost is equal to marginal cost, we need to determine the derivative of the average cost function and equate it to the marginal cost.

First, let's find the average cost function by dividing the total cost by the quantity of output (x):

Average cost = (1/3 x^3 – 18x^2 + 160x) / x

Simplifying this expression, we get:

Average cost = (1/3 x^2 – 18x + 160)

Next, let's find the derivative of the average cost function:

d(Average cost) / dx = d/dx (1/3 x^2 – 18x + 160)

d(Average cost) / dx = (2/3)x - 18

Now, let's equate the derivative of the average cost (marginal cost) to the marginal cost itself:

(2/3)x - 18 = marginal cost

Since we're looking for the output at which average cost is equal to marginal cost, we need to solve this equation for x.

Adding 18 to both sides, we get:

(2/3)x = marginal cost + 18

Multiplying both sides by 3/2:

x = (3/2) * (marginal cost + 18)

Therefore, the output at which average cost is equal to marginal cost is given by: x = (3/2) * (marginal cost + 18)

Answer 2

To find the output at which average cost is equal to marginal cost, we need to understand the concepts of average cost and marginal cost.

Average cost is the cost per unit of output and is calculated by dividing the total cost by the number of units produced. Mathematically, average cost (AC) is given by the formula:

AC = Total Cost / Number of Units

Marginal cost, on the other hand, is the additional cost incurred to produce one more unit of output. It is the derivative of the total cost function with respect to the number of units. Mathematically, marginal cost (MC) is given by the formula:

MC = d(Total Cost) / d(Number of Units)

Now, let's calculate the average cost and marginal cost for the given total cost function:

Total Cost = 1/3 x^3 – 18x^2 + 160x

To find the average cost, we divide the total cost by the number of units produced, which is x. So,

AC = Total Cost / x

To find the marginal cost, we take the derivative of the total cost function with respect to x:

MC = d(Total Cost) / dx

Now, differentiate the total cost function:

MC = (d/dx)(1/3 x^3 – 18x^2 + 160x)

MC = (1/3) * 3x^2 - 2 * 18x + 160

MC = x^2 - 36x + 160

To find the output at which average cost is equal to marginal cost, we need to set AC equal to MC:

AC = MC

Total Cost / x = x^2 - 36x + 160

Simplify the equation:

1/3 x^3 – 18x^2 + 160x = x^3 - 36x^2 + 160x

1/3 x^3 = x^3

1/3 = 1

Since the equation is not true for any value of x, there is no output at which the average cost is equal to the marginal cost for this specific total cost function.

Answer 3

average cost is

(1/3 x^3 – 18x^2 + 160x)/x
= 1/3 x^2 - 18x + 160

marginal cost is

x^2-26x+160

so, where is

1/3 x^2 - 18x = x^2 - 26x ?