Question

$N$ characters of information are held on magnetic tape, in batches of $x$ characters each; the batch processing time is $\alpha +\beta x^2$ seconds; $\alpha ,\beta$ are constants. The optimum value of $x$ for fast processing is

• A. $\frac{\alpha }{\beta }$
• B. $\frac{\beta }{\alpha }$
• C. $\sqrt{\frac{\alpha }{\beta }}$
• D. $\sqrt{\frac{\beta }{\alpha }}$

Answer 1

Answer: (C)

$\sqrt{\frac{\alpha }{\beta }}$

$T=\frac{N}{x}(\alpha +\beta x^2)=N(\frac{\alpha }{x}+\beta x)\Rightarrow \frac{dT}{dx}=N(-\frac{\alpha }{x^2}+\beta )$

$\therefore \frac{dT}{dx}=0\Rightarrow x^2=\frac{\alpha }{\beta }\Rightarrow x=\sqrt{\frac{\alpha }{\beta }}$

Also, $\frac{d^{2}T}{{dx}^2}=N\left(\frac{2\alpha }{x^3}\right)>0$ at $x=\sqrt{\frac{\alpha }{\beta }}$

$\therefore T$ is minimum for $x=\sqrt{\frac{\alpha }{\beta }}$