Applications Of Derivatives Question 231
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 optimal value of x for fast processing is
[MNR 1986]
Options:
A) $ \frac{\alpha }{\beta } $
B) $ \frac{\beta }{\alpha } $
C) $ \sqrt{\frac{\alpha }{\beta }} $
D) $ \sqrt{\frac{\beta }{\alpha }} $
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Answer:
Correct Answer: C
Solution:
Here number of batches $ =\frac{N}{x} $ and time per batch $ =(\alpha +\beta x^{2}), $ second
$ \therefore $ Total processing time $ T=( \frac{N}{x} ),(\alpha +\beta x^{2})=N( \frac{\alpha }{x}+\beta x )second $
For fast processing T must be least,
$ \therefore \frac{dT}{dx}=N( -\frac{\alpha }{x^{2}}+\beta ),;\ \ \frac{d^{2}T}{dx^{2}}=\frac{2N\alpha }{x^{3}} $
For maxima or minima of $ T,\ \ \frac{dT}{dx}=0\Rightarrow x=\sqrt{( \frac{\alpha }{\beta } )} $
For $ x=\sqrt{( \frac{\alpha }{\beta } )},\frac{d^{2}T}{dx^{2}} $ is + $ ve\ \ i.e.,>0 $
$ \therefore $ T has minima for $ x=\sqrt{( \frac{\alpha }{\beta } )} $ .
BETA
