Two sound waves of frequencies 100 Hz and 102 Hz and having same amplitude 'A' are interfering. A detector which can detect waves of amplitude greater than or equal to A is kept at rest. In a time interval of 12 seconds, find the total duration (in sec) in which detector is active.

${\text{y}}_{\text{1}}{\text{ = A sin w}}_{\text{1}}\text{t}$ and ${\text{y}}_{\text{2}}{\text{ = A sin w}}_{\text{2}}\text{t}$ 
$y_r=2A\cos \left\{\frac{({\omega }_2-{\omega }_1)}{2}t\right\}\left\{\sin \frac{({\omega }_2+{\omega }_1)}{2}t\right\}$
Hence Resultant amplitude $A_r=2A_0|\cos \left(\Delta \omega \right)t/2|$
$\Delta \omega \frac{t}{2}=\frac{\pi }{2}\Rightarrow t=\frac{1}{4}\sec$
$\Delta \omega \frac{t}{2}=\frac{\pi }{3}\Rightarrow t=\frac{1}{6}\sec$
In one cycle of intensity of 1/2s, the detector remains idle for $2(\frac{1}{4}-\frac{1}{6})s=\frac{1}{6}\sec$
Hence in 18 seconds detector remains active for $12\frac{(1/3)}{(1/2)}=8\sec$