Question

A particle is moving in a straight line according to the formula s = $t^2$ + 8t + 12. If s be measured in metre and t be measured in second, then the average velocity of the particle in third second is

• A. 14 m / sec
• B. 13 m / sec
• C. 15 m / sec
• D. None of these

Answer 1

The correct option is C 15 m / sec

Here number of batches $=\frac{N}{x}$ and time per batch $(\alpha =\beta x^2)$ second

$\therefore$ Total processing time $T=\left(\frac{N}{x}\right)(\alpha +\beta x^2)=N(\frac{\alpha }{x}+\beta x)$ second

For fast processing T must be least,
$\therefore \frac{dT}{dx}=N(-\frac{\alpha }{x^2+\beta });\frac{d^{2}T}{d{x}^2}=\frac{2N\alpha }{x^3}$

For maxima or minima of T, $\frac{dT}{dx}=0\Rightarrow x=\sqrt{\left(\frac{\alpha }{\beta }\right)}$

For x = $\sqrt{\left(\frac{\alpha }{\beta }\right)},\frac{d^{2}T}{d{x}^2}$is + ve i.e., > 0

$\therefore$T has minima for $x=\sqrt{\frac{\alpha }{\beta }}.$