The cost of manufacturing of certain items consists of ₹1600 as overheads, ₹30 per item as the cost of material and the labour cost $\frac{x^2}{100}$ for x items produced. How many items must be produced to have a minimum average cost?

400

The correct answer is Option (4) → 400

The cost of material for producing x items = $₹30x$.

Overhead cost = ₹1600 and labour cost of producing x items = $₹\frac{x^2}{100}$.

∴ Total cost of producing x items, $C(x) =₹\left(1600 + 30x +\frac{x^2}{100}\right)$.

∴ Average cost, $AC =\frac{C(x)}{x}=₹\left(\frac{1600}{x}+30+\frac{x}{100}\right)$.

To find value(s) of x so that AC is minimum, we should find value(s) of x where $\frac{d}{dx}(AC)=0$ and $\frac{d^2}{dx^2}(AC) > 0$.

Now, $\frac{d}{dx}(AC)=-\frac{1600}{x^2}+0+\frac{1}{100}$ and $\frac{d^2}{dx^2}(AC)=-1600 (−2)x^{−3} + 0 = \frac{3200}{x^3}$.

$\frac{d}{dx}(AC)=0⇒-\frac{1600}{x^2}+\frac{1}{100}=0⇒x^2=160000$

$⇒x = 400, -400$ but $x > 0$

$⇒x = 400$.

When $x=400,\frac{d^2}{dx^2}(AC)=\frac{3200}{(4000)^3}>0$

Hence, the average cost is minimum when $x = 400$.