A man in a car at location \(Q\) on a straight highway is moving with speed \(v\). He decides to reach a point \(P\) in a field at a distance \(d\) from highway (point \(M\) ) as shown in the figure.Speed of the car in the field is half to that on the highway. What should be the distance \(RM\), so that the time taken to reach \(P\) is minimum?

\(\begin{array}{l}
Let\,the\,car\,turn\,of\,the\,highway\,at\,a\,{\rm{distance }}\\
'x'\,from\,the\,{\rm{point}}\,M.\,So,\,RM = x\\
And\,if\,speed\,of\,car\,in\,field\,is\,v,\,then\,time\,\\
taken\,by\,the\,car\,to\,{\rm{cover}}\,{\rm{the}}\,{\rm{distance}}\,{\rm{QR = }}\\
{\rm{QM}}\, - {\rm{x}}\,{\rm{on}}\,the\,highway,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,{t_1} = \frac{{QM - x}}{{2v}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)\\
Time\,taken\,to\,travel\,the\,{\rm{distance}}\,'R\,P'\,in\,the\\
filed\\
\,\,\,\,\,\,\,\,\,\,\,\,\,{t_2} = \frac{{\sqrt {{d^2} + {x^2}} }}{v}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....\left( {ii} \right)
\end{array}\)

\(\begin{array}{l}
Total\,time\,elapsed\,to\,move\,the\,car\,form\,Q\\
to\,P\\
t = {t_1} + {t_2} = \frac{{QM - x}}{{2v}} + \frac{{\sqrt {{d^2} + {x^2}} }}{v}\,\,\,\,\,\\
For't'\,to\,be\,mimimum\,\frac{{dt}}{{dx}} = 0\\
\frac{1}{v}\left[ { - \frac{1}{2} + \frac{x}{{\sqrt {{d^2} + {x^2}} }}} \right] = 0\\
\,\,\,\,\,\,\,\,\,\,\,or\,x = \frac{d}{{\sqrt {{2^2} - 1} }} = \frac{d}{{\sqrt 3 }}
\end{array}\)