Question

Two roads OA and OB intersects at an angle of ${60}^o$. A car driver approaches O from A where OA = 800 m at a uniform speed of 20m/s. Simultaneously another car moves from O towards B at a uniform speed of 25m/sec. If t is the time when two cars are closest, then $\left[\frac{t}{2}\right]$ is equal to (where [ . ] denotes G.I.F).

Answer 1

Let the dis. b/w two cars after $t$ second be $S$. Distance covered in $t$ is $20t$, so its distance from $O$ will be $(800-20t)m$

Then the distance covered by second car is $20tm$.

$\frac{1}{2}=\cos {60}^o=\frac{(800-20t{)}^2+(25t{)}^2-S^2}{(800-20t)25t}$

$S^2=(800-20t{)}^2+625t^2-(40-t)250t$

for $S$ to be min, we have $\frac{ds}{dt}=0$

$-40(800-20t)+1250t-(40-t)250+250t=0$

$-42000+2550t=0$

$t=\frac{840}{51}Sr$

$\frac{t}{2}=\frac{420}{51}$

$\left[\frac{t}{2}\right]=\left[8.23\right]$

$=8Sr$