Question

N characters of information are held on a magnetic tape in batches of x characters each, the batch processing time being $(4+9x^2)$ seconds. The optimal value of x for fast processing is

• A. $\frac{4}{9}$
• B. $\frac{9}{4}$
• C. $\frac{2}{3}$
• D. $\frac{3}{2}$

Answer 1

Answer: (C)

$\frac{2}{3}$

Number of batches $=\frac{N}{x}$
Processing time for 1 batch $=4+9x^2$
So, total processing time $=\frac{N}{x}(4+9x^2)$
Let $f\left(x\right)=\frac{N}{x}(4+9x^2)$
$f^{\prime }\left(x\right)=\frac{N(9x^2-4)}{x^2}$
For maxima or minima,
$f^{\prime }\left(x\right)=0$
$\Rightarrow x=\pm \frac{2}{3}$
$f^{\prime \prime }\left(x\right)=\frac{8}{x^3}$
$f^{\prime \prime }\left(x\right)>0forx=\frac{2}{3}$
Hence , the minimum value of x for fast processing is $\frac{2}{3}$