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# Two sides of a tringle have length $12$m and $15$m. The angle between them is increasing at a rate of $2^{\circ}/min.$ How fast is the length of third side increasing when the angle between the sides of fixe is $60^{\circ}$?

Toolbox:
• If $s=f(t)$ is the distance function, representing the distance 's' travelled by a particle in time t, then the velocity and acceleration functions are $v=\large\frac{ds}{dt}$$=t'(t) and a=\large\frac{d^2s}{dt^2}$$=f''(t)$
• It $y=f(x)$ and both depend on 't' then $\large\frac{dy}{dt}$$=f'(x).\large\frac{dx}{dt} Let OA,OB be the sides of \Delta OAB where OA=12\;m,OB=15\; m and \angle AOB=\theta at time t The length h of the third side AB at time t is given by s^2=OA^2+OB^2-2OA-OB\cos \theta s^2=144+225-360\;\cos \theta \quad= 369-360 \;\cos \theta-----(i) Differentiating (i) with respect to 't' we have 2s \large\frac{ds}{dt}$$=+360 \sin \theta \large\frac{d\theta}{d t}$
$=>\large\frac{ds}{dt}=\large\frac{180}{s} $$\sin \theta \large\frac{d \theta}{dt}-----(ii) When \theta=60 ^{\circ}=\large \frac{\pi ^c}{3} \large\frac{d\theta}{dt}$$=2^{\circ}/min=\large\frac{\pi^c}{90}$$/min and s^2=369-360 \times \large\frac{1}{2}$$=189$
ie $s=\sqrt {189}=3 \sqrt {21}$
At this instant substituting for $s, \large\frac{d \theta}{dt}$ in (ii)
$\large\frac{ds}{dt}=\large \frac{+180}{3 \sqrt {21}}\;$$\sin \large\frac{\pi}{3} \times \frac{\pi}{90} \qquad=\large\frac{+60}{\sqrt {21}} \times \frac{\sqrt 3}{2} \times \frac{\pi}{90} \qquad=\large\frac{+ \sqrt 3 \pi}{3 \sqrt {21}}=\frac{+\pi}{\sqrt {63}}$$m/sec$