Question

Minimum Average Cost

The cost of producing x units of a product is modeled by
C = 500 + 300x - 300 ln x, x ≥ 1.
(a) Find the average cost function Ć.
(b) Find the minimum average cost analytically. Use a graphing utility to confirm your result.

Answer 1

a)
`\overline{C(x)} = 500x^(-1) + 300 - 300\frac{\ln({x})}{x}`

b)
`\overline{C(e^{(8)/(3)})} = 279.1549`

Explanation:

Given the cost function as C(x):

`C(x) = 500 + 300x - 300ln(x) \quad\quad,x\geq1`

a) Find the average cost function,
`(\overline{C(x)})`

if C is the cost of selling x units, The Average can be denoted by:

`\overline{C(x)} = \frac{\text{total cost of selling x units}}{\text{x units}}`

`\overline{C(x)} = (C(x))/(x)`

`\overline{C(x)} = (500 + 300x - 300ln((x)))/(x)`

`\overline{C(x)} = 500x^(-1) + 300 - 300\frac{\ln({x})}{x}`

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
`(\overline{C(x)})`. and equate it to zero. (like finding the stationary point of any function)

`(d)/(dx)(\overline{C(x)}) = (d)/(dx)\left(500x^(-1) + 300 - 300\frac{\ln({x})}{x}\right)`

`\overline{C'(x)} = -500x^(-2) + 0 - 300(x(1)/(x) - ln((x)))/(x^2)}`

now just simplify:

`\overline{C'(x)} = -(500)/(x^2)-300(1 - ln((x)))/(x^2)}`

`\overline{C'(x)} = -(1)/(x^2)(500+300(1 - ln((x))))`

we've found the derivative of C(x), now to find the minimum we'll equate this derivative to zero.
`\overline{C'(x)} = 0`

`0 = -(1)/(x^2)(500+300(1 - ln((x))))`

and now solve for x

`0 = 500+300(1 - ln((x)))`

`-500 = 300(1 - ln((x)))`

`1 + (5)/(3)=ln((x))`

`ln((x))=(8)/(3)`

`x=e^{(8)/(3)}\approx 14.392`

at this value of x the average cost is minimum.

`\overline{C(x)} = 500x^(-1) + 300 - 300\frac{\ln({x})}{x}`

`\overline{C(e^{(8)/(3)})} = 500{e^{-(8)/(3)}} + 300 - 300\frac{\ln({e^{(8)/(3)}})}{e^{(8)/(3)}}`

`\overline{C(e^{(8)/(3)})} = 500{e^{-(8)/(3)}} + 300 - 300\frac{8}{3e^{(8)/(3)}}}`

`\overline{C(e^{(8)/(3)})} = 34.7417 + 300 -55.5868`

`\overline{C(e^{(8)/(3)})} = 279.1549`

This is the minimum average cost!