Answer:
a)[tex]\overline{C(x)} = 500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}[/tex]
b)[tex]\overline{C(e^{\frac{8}{3}})} = 279.1549[/tex]
Step-by-step explanation:
Given the cost function as C(x):
[tex]C(x) = 500 + 300x - 300\ln{x} \quad\quad,x\geq1[/tex]
a) Find the average cost function, [tex](\overline{C(x)})[/tex]
if C is the cost of selling x units, The Average can be denoted by:
[tex]\overline{C(x)} = \dfrac{\text{total cost of selling x units}}{\text{x units}}[/tex]
[tex]\overline{C(x)} = \dfrac{C(x)}{x}[/tex]
[tex]\overline{C(x)} = \dfrac{500 + 300x - 300\ln{(x)}}{x}[/tex]
[tex]\overline{C(x)} = 500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}[/tex]
this is the average cost function
b) The minimum average cost:
To find the minimum average cost, we'll have to differentiate the average cost function [tex](\overline{C(x)})[/tex]. and equate it to zero. (like finding the stationary point of any function)
[tex]\dfrac{d}{dx}(\overline{C(x)}) = \dfrac{d}{dx}\left(500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}\right)[/tex]
[tex]\overline{C'(x)} = -500x^{-2} + 0 - 300\dfrac{x\frac{1}{x} - \ln{(x)}}{x^2}}[/tex]
now just simplify:
[tex]\overline{C'(x)} = -\dfrac{500}{x^2}-300\dfrac{1 - \ln{(x)}}{x^2}}[/tex]
[tex]\overline{C'(x)} = -\dfrac{1}{x^2}(500+300(1 - \ln{(x)}))[/tex]
we've found the derivative of C(x), now to find the minimum we'll equate this derivative to zero. [tex]\overline{C'(x)} = 0[/tex]
[tex]0 = -\dfrac{1}{x^2}(500+300(1 - \ln{(x)}))[/tex]
and now solve for x
[tex]0 = 500+300(1 - \ln{(x)})[/tex]
[tex]-500 = 300(1 - \ln{(x)})[/tex]
[tex]1 + \dfrac{5}{3}=\ln{(x)}[/tex]
[tex]\ln{(x)}=\dfrac{8}{3}[/tex]
[tex]x=e^{\frac{8}{3}}\approx 14.392[/tex]
at this value of x the average cost is minimum.
[tex]\overline{C(x)} = 500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}[/tex]
[tex]\overline{C(e^{\frac{8}{3}})} = 500{e^{-\frac{8}{3}}} + 300 - 300\dfrac{\ln({e^{\frac{8}{3}}})}{e^{\frac{8}{3}}}[/tex]
[tex]\overline{C(e^{\frac{8}{3}})} = 500{e^{-\frac{8}{3}}} + 300 - 300\dfrac{8}{3e^{\frac{8}{3}}}}[/tex]
[tex]\overline{C(e^{\frac{8}{3}})} = 34.7417 + 300 -55.5868 [/tex]
[tex]\overline{C(e^{\frac{8}{3}})} = 279.1549[/tex]
This is the minimum average cost!
