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:
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)
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.
18x
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 ?