when no more than 110 units are produced, the cost of producing x units is given by
C(x)=.2x^3 - 25x^2 + 1531x + 25,000
How many units should be produced in order to have the lowest possible average cost?
C '(x) = .6x^2 - 50x + 1531
= 0 for a max/min of C
solve using the quadratic formula
This quadratic has no real roots, I will leave it up to you to interpret that result.
may i ask how you came up with this equation? thank you
I took the derivative.
Is this not from a Calculus course ?
no its not. its from a business math course.but i think they're a pretty similar course?
To find the number of units that should be produced in order to have the lowest possible average cost, we need to find the minimum value of the average cost function.
The average cost is calculated by dividing the total cost by the number of units produced. In this case, the total cost is given by C(x) and the number of units produced is x. Therefore, the average cost function can be represented as:
AC(x) = (C(x))/x
To find the minimum value of AC(x), we need to find the critical points. The critical points occur when the derivative of AC(x) is equal to zero or does not exist. So, let's differentiate AC(x) with respect to x:
AC'(x) = (C'(x) * x - C(x)) / x^2
Now, let's find C'(x) by differentiating C(x) with respect to x:
C'(x) = 0.6x^2 - 50x + 1531
Substituting C'(x) and C(x) into AC'(x), we get:
AC'(x) = ((0.6x^2 - 50x + 1531) * x - (.2x^3 - 25x^2 + 1531x + 25,000)) / x^2
Simplifying further:
AC'(x) = (0.6x^3 - 50x^2 + 1531x^2 - 0.2x^3 + 25x^2 - 1531x - 25,000) / x^2
AC'(x) = (0.4x^3 + 6x^2 - 1531x - 25,000) / x^2
Now, to find the critical points, we need to set AC'(x) equal to zero and solve for x:
0 = (0.4x^3 + 6x^2 - 1531x - 25,000) / x^2
Multiplying both sides by x^2:
0 = 0.4x^3 + 6x^2 - 1531x - 25,000
We can solve this equation to find the critical points. However, since this is a cubic equation, it may not have a simple algebraic solution. Therefore, we can use numerical methods such as graphical analysis or the Newton-Raphson method to find the values of x that minimize the average cost.