Question

The cost of manufacturing of certain items consists of ₹ 1,600 as overheads, ₹ 30 per item as the cost of the material and the labour cost ₹ x²/100 for x items produced.

How many items must be produced to have a minimum average cost?

Answer 1

Cost function (C) = \(\frac{x^2}{100}+30x+1600\)

Average cost (A.C) = \(\frac{C}{x}=\frac{x}{100}+30+\frac{1600}{x}\)

Now,

\(\frac{d}{dx}(A.C)=\frac{1}{100}+0-\frac{1600}{x^2}\)

For maximum and minimum cost d/dx (A.C) = 0

\(\therefore \,\frac{1}{100}-\frac{1600}{x^2}=0\)

\(⇒\frac{1600}{x^2}=\frac{1}{100}\)

⇒ x² = 160000

⇒ x = 400

\(\frac{d^2(A.C.)}{dx^2}=\frac{3200}{x^3}=+ve,\) at x = 400

∴ A.C. is minimum at x = 400.

∴ 400 items must be produced to have a minimum average cost.