The cost of producing x units of a product is given by C(x)=600+90x−90ln(x),x≥1. Find the minimum average cost.

Question

The cost of producing x units of a product is given by
C(x)=600+90x−90ln(x),x≥1.

Find the minimum average cost.

Answer 1

Average Cost Of Production Per Unit or C-bar is given by Total Cost For Production Of X Units Divided By X Units, which is written as (600 + 90x − 90ln x)/x equal to 600x^-1 + 90 − (90ln x)/x.

To minimize this Average Cost, differentiate C-bar; d(C-bar)/dx = d(600x^-1 + 90 − (90ln x)/x)/dx, which gives

-600x^-2 + 0 − [x(90/x) − (90ln x)(1)]/x². This simplifies to -600x^-2 − 90x^-2 + x^-2(90ln x) or d(C-bar)/dx =

(90ln x− 690)/x².

Solve (90ln x− 690)/x² = 0 to obtain ln x = (690/90) or x = e^(690/90) or 2135.949733.

Next take the derivative of (90ln x− 690)/x² : [x²(90/x) − (90ln x− 690)2x]/x⁴ which reduces to

[90 − 180ln x + 1380]/x³ or [1470 −180ln x]/x³. [1470 −180ln x]/x³ is greater than 0 at x = 2135.949733 which means that Average Cost (C-bar) has a minimum at x = 2135.949733. This minimum of Average Cost (C-bar) is equal to [600 + 90(2135.949733) − 90ln (2135.949733)]/2135.949733 or 89.95786419 or $89.96.