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,$\alpha and\beta$ are constants. The optical value of x for fast processing is,

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

Answer 1

Answer: (C)

$\sqrt{\alpha /\beta }$

Number of batches $=\frac{N}{x}$ and time per batch $=(\alpha +\beta x^2)$ seconds

Hence total time T is given by $T=\frac{N}{x}(\alpha +\beta x^2)=N(\frac{\alpha }{x}+\beta x)$ seconds.

For fast processing T must be least.Now $\frac{dT}{dx}=N(-\frac{\alpha }{x^2}+\beta )=0$ for max.or min.

or $x=\sqrt{\left(\frac{\alpha }{\beta }\right)}$ and $\frac{d^{2}T}{d{x}^2}=N.\frac{2\alpha }{x^3}>0$

$\therefore T$ is least when $x=\sqrt{\left(\frac{\alpha }{\beta }\right)}.$

Ans: C