Here's one more that I do not understand
A company estimates that the cost ($) of producing x units of a certain product is given by:
C= 800 + .04x + .0002x^2
Find the production level that minimizes the average cost per unit.
Thank you so much
your question makes no sense.
clearly C is increasing as x increases, and C will have a minimum of 800 when x = 0
Their best bet would be not to make any ???
check your typing, I think it should be -.002x^2 perhaps?
Sorry for the typo - you are correct it should be -.002x^2
in that case, very easy
dC/dx = .04 - .004x
= 0 and solve for x
x = .004/.04 = ...
I still do not see what the answer would be . Is x = .001
To find the production level that minimizes the average cost per unit, we need to first find the average cost function and then determine its minimum.
The average cost per unit can be found by dividing the total cost (C) by the number of units produced (x).
Total cost (C) is given by the equation:
C(x) = 800 + 0.04x + 0.0002x^2
And the number of units produced is denoted by x.
Average cost (AC) is calculated by dividing the total cost (C) by the number of units produced (x):
AC(x) = C(x) / x
Substituting the equation for C(x), we get:
AC(x) = (800 + 0.04x + 0.0002x^2) / x
To find the production level that minimizes the average cost per unit, we need to find the minimum point of the average cost function. To do this, we can take the derivative of AC(x) with respect to x and set it equal to zero.
The derivative of AC(x) can be found using the quotient rule:
AC'(x) = (800 + 0.04x + 0.0002x^2)' / x - (800 + 0.04x + 0.0002x^2) / x^2
Simplifying the derivative expression, we get:
AC'(x) = (0.04 + 0.0004x - 0.0002x) / x - (800 + 0.04x + 0.0002x^2) / x^2
Now, set AC'(x) equal to zero and solve for x:
0 = (0.04 + 0.0004x - 0.0002x) / x - (800 + 0.04x + 0.0002x^2) / x^2
0 = 0.04 + 0.0004x - 0.0002x - (800 + 0.04x + 0.0002x^2) / x^2
Now, multiply both sides of the equation by x^2 to eliminate the denominator:
0 = 0.04x^2 + 0.0004x^3 - 0.0002x^3 - (800x + 0.04x^2 + 0.0002x^3)
0 = 0.04x^2 + 0.0004x^3 - 0.0002x^3 - 800x - 0.04x^2 - 0.0002x^3
Simplifying further:
0 = -800x
Now, solve for x:
x = 0
Therefore, the production level that minimizes the average cost per unit is x = 0.
However, it is important to note that this result might not make practical sense depending on the context of the problem. In most cases, a production level of zero would not be meaningful. Therefore, it is recommended to check if there are any other relevant restrictions or conditions specified in the problem statement.