Question

The manufacturing cost of an item consists ofRs900 as overheads. The material costs Rs3 per item and labor cost Rs x²/100 for x items produced. how many items must be produced to have minimum average cost?

Answer 1

Dear Student,

Let x items be produced
Total Cost of x items = 3x + $\frac{{\mathrm{x}}^2}{100}$ + 900
Average Cost (C) = $\frac{3\mathrm{x}+{\displaystyle \frac{{\mathrm{x}}^2}{100}}+900}{\mathrm{x}}$ = $3+\frac{\mathrm{x}}{100}+\frac{900}{\mathrm{x}}$
$\frac{d\mathrm{C}}{d\mathrm{x}}$ = $\frac{1}{100}-\frac{900}{{\mathrm{x}}^2}$

$$\begin{gathered} \mathrm{For}\mathrm{maximum}\mathrm{or}\mathrm{minimum}, \\ \frac{\mathrm{dC}}{\mathrm{dx}}=0 \\ \Rightarrow \frac{1}{100}-\frac{900}{{\mathrm{x}}^2}=0 \\ \Rightarrow {\mathrm{x}}^2=90000 \\ \Rightarrow \mathrm{x}=300 \end{gathered}$$



$$\begin{gathered} \frac{{\mathrm{d}}^2\mathrm{C}}{{\mathrm{dx}}^2}=\frac{1800}{{\mathrm{x}}^3} \\ \mathrm{Now},{\left[\frac{{\mathrm{d}}^2\mathrm{C}}{{\mathrm{dx}}^2}\right]}_{\mathrm{x}=300}=\frac{1800}{{\left(300\right)}^3}>0 \end{gathered}$$

So, 300 items should be produced to minimise the average cost.

Regards