The cost of producing x units of a product is given by C(x)=700+ 80x - 80ln(x), x is greater or equal to 1. Find the minimum average cost.
the average cost is C(x)/x = 700/x + 80 - 80lnx/x
C'(x) = 20/x^2 (4lnx - 39)
C' = 0 at x = e^(39/4) ≈ 17154
C(17154) = 80
C ' (x) = 80 - 80(1/x) = 0 for a min
80 = 80/x
1 = 1/x
x = 1
state the conclusion
To find the minimum average cost, we need to first calculate the average cost function. The average cost is the total cost divided by the number of units produced.
The total cost function is given by the equation C(x) = 700 + 80x - 80ln(x).
The number of units produced is denoted by x.
To find the average cost, we divide the total cost function by the number of units produced, which is x. Therefore, the average cost function, denoted by AC(x), is given by AC(x) = C(x)/x.
Substituting the total cost function into the average cost function, we have:
AC(x) = (700 + 80x - 80ln(x))/x.
To find the minimum average cost, we need to find the critical points of the average cost function. We do this by finding the derivative of the average cost function and setting it equal to zero.
Taking the derivative of AC(x) with respect to x, we can use the quotient rule:
AC'(x) = [(x) * (80) - (700 + 80x - 80ln(x))] / (x^2).
Simplifying the numerator, we have:
AC'(x) = (80x - 700 - 80x + 80ln(x)) / (x^2).
Combining like terms, we get:
AC'(x) = (80ln(x) - 700) / (x^2).
Setting this derivative equal to zero:
(80ln(x) - 700) / (x^2) = 0.
To solve this equation, we need to multiply both sides by (x^2):
80ln(x) - 700 = 0.
Adding 700 to both sides, we have:
80ln(x) = 700.
Dividing both sides by 80, we get:
ln(x) = 700 / 80.
Taking the exponent of both sides, we have:
x = e^(700/80).
Calculating this value, we find:
x ≈ 17.6.
So the critical point of the average cost function is x = 17.6.
Next, we need to check if this critical point is the minimum by analyzing the second derivative. If the second derivative is positive at x = 17.6, then it is a minimum. Otherwise, it is not.
To find the second derivative of AC(x), we differentiate AC'(x):
AC''(x) = 80 / (x^2) - (80ln(x) - 700) / (x^3).
Simplifying, we have:
AC''(x) = (80 - 80ln(x) + 700) / (x^3).
Substituting x = 17.6 into this second derivative expression, we find:
AC''(17.6) ≈ 1.45.
Since AC''(17.6) is positive, it means that x = 17.6 is the point of minimum average cost.
Therefore, the minimum average cost is given by AC(17.6).
To find this value, substitute x = 17.6 into the average cost function:
AC(17.6) = (700 + 80(17.6) - 80ln(17.6))/17.6.
Evaluate the expression to find the minimum average cost.